diff options
Diffstat (limited to '')
56 files changed, 8839 insertions, 27 deletions
diff --git a/buch/papers/reedsolomon/.gitignor b/buch/papers/reedsolomon/.gitignor new file mode 100644 index 0000000..52a02ac --- /dev/null +++ b/buch/papers/reedsolomon/.gitignor @@ -0,0 +1,24 @@ +RS.aux +RS.bbl +RS.bib +RS.blg +RS.idx +RS.ilg +RS.ind +RS.log +RS.out +RS.pdf +RS.run.xml +RS.toc +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb diff --git a/buch/papers/reedsolomon/RS presentation/README.txt b/buch/papers/reedsolomon/RS presentation/README.txt new file mode 100644 index 0000000..4d0620f --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/README.txt @@ -0,0 +1 @@ +Dies ist die Presentation des Reed-Solomon-Code
\ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.aux b/buch/papers/reedsolomon/RS presentation/RS.aux new file mode 100644 index 0000000..065ba66 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.aux @@ -0,0 +1,146 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand*\HyPL@Entry[1]{} +\bbl@beforestart +\catcode `"\active +\HyPL@Entry{0<</P(1)>>} +\babel@aux{ngerman}{} +\@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {1}{1}}} +\HyPL@Entry{1<</P(2)>>} +\@writefile{toc}{\beamer@sectionintoc {1}{Einführung}{2}{0}{1}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {1}{1}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{1}}} +\@writefile{nav}{\headcommand {\sectionentry {1}{Einführung}{2}{Einführung}{0}}} +\@writefile{nav}{\headcommand {\slideentry {1}{0}{1}{2/4}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {2}{4}}} +\HyPL@Entry{4<</P(3)>>} +\@writefile{toc}{\beamer@sectionintoc {2}{Polynom Ansatz}{5}{0}{2}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {2}{4}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {2}{4}}} +\@writefile{nav}{\headcommand {\sectionentry {2}{Polynom Ansatz}{5}{Polynom Ansatz}{0}}} +\@writefile{nav}{\headcommand {\slideentry {2}{0}{1}{5/5}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {5}{5}}} +\HyPL@Entry{5<</P(4)>>} +\@writefile{nav}{\headcommand {\slideentry {2}{0}{2}{6/8}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {6}{8}}} +\HyPL@Entry{8<</P(5)>>} +\@writefile{nav}{\headcommand {\slideentry {2}{0}{3}{9/12}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {9}{12}}} +\HyPL@Entry{12<</P(6)>>} +\@writefile{toc}{\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{13}{0}{3}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {5}{12}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {5}{12}}} +\@writefile{nav}{\headcommand {\sectionentry {3}{Diskrete Fourier Transformation}{13}{Diskrete Fourier Transformation}{0}}} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{1}{13/13}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {13}{13}}} +\HyPL@Entry{13<</P(7)>>} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{2}{14/20}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {14}{20}}} +\HyPL@Entry{20<</P(8)>>} +\@writefile{snm}{\beamer@slide {ft_discrete}{21}} +\newlabel{ft_discrete}{{8}{21}{Diskrete Fourier Transformation}{Doc-Start}{}} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{3}{21/23}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {21}{23}}} +\HyPL@Entry{23<</P(9)>>} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{4}{24/24}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {24}{24}}} +\HyPL@Entry{24<</P(10)>>} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{5}{25/26}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {25}{26}}} +\HyPL@Entry{26<</P(11)>>} +\@writefile{toc}{\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{27}{0}{4}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {13}{26}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {13}{26}}} +\@writefile{nav}{\headcommand {\sectionentry {4}{Reed-Solomon in Endlichen Körpern}{27}{Reed-Solomon in Endlichen Körpern}{0}}} +\@writefile{nav}{\headcommand {\slideentry {4}{0}{1}{27/27}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {27}{27}}} +\HyPL@Entry{27<</P(12)>>} +\@writefile{nav}{\headcommand {\slideentry {4}{0}{2}{28/28}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {28}{28}}} +\HyPL@Entry{28<</P(13)>>} +\@writefile{toc}{\beamer@sectionintoc {5}{Codierung eines Beispiels}{29}{0}{5}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {27}{28}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {27}{28}}} +\@writefile{nav}{\headcommand {\sectionentry {5}{Codierung eines Beispiels}{29}{Codierung eines Beispiels}{0}}} +\@writefile{nav}{\headcommand {\slideentry {5}{0}{1}{29/29}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {29}{29}}} +\HyPL@Entry{29<</P(14)>>} +\@writefile{nav}{\headcommand {\slideentry {5}{0}{2}{30/30}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {30}{30}}} +\HyPL@Entry{30<</P(15)>>} +\@writefile{toc}{\beamer@sectionintoc {6}{Decodierung ohne Fehler}{31}{0}{6}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {29}{30}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {29}{30}}} +\@writefile{nav}{\headcommand {\sectionentry {6}{Decodierung ohne Fehler}{31}{Decodierung ohne Fehler}{0}}} +\@writefile{nav}{\headcommand {\slideentry {6}{0}{1}{31/31}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {31}{31}}} +\HyPL@Entry{31<</P(16)>>} +\@writefile{nav}{\headcommand {\slideentry {6}{0}{2}{32/34}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {32}{34}}} +\HyPL@Entry{34<</P(17)>>} +\@writefile{nav}{\headcommand {\slideentry {6}{0}{3}{35/35}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {35}{35}}} +\HyPL@Entry{35<</P(18)>>} +\@writefile{toc}{\beamer@sectionintoc {7}{Decodierung mit Fehler}{36}{0}{7}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {31}{35}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {31}{35}}} +\@writefile{nav}{\headcommand {\sectionentry {7}{Decodierung mit Fehler}{36}{Decodierung mit Fehler}{0}}} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{1}{36/36}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {36}{36}}} +\HyPL@Entry{36<</P(19)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{2}{37/37}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {37}{37}}} +\HyPL@Entry{37<</P(20)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{3}{38/38}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {38}{38}}} +\HyPL@Entry{38<</P(21)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{4}{39/39}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {39}{39}}} +\HyPL@Entry{39<</P(22)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{5}{40/40}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {40}{40}}} +\HyPL@Entry{40<</P(23)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{6}{41/42}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {41}{42}}} +\HyPL@Entry{42<</P(24)>>} +\@writefile{toc}{\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{43}{0}{8}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {36}{42}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {36}{42}}} +\@writefile{nav}{\headcommand {\sectionentry {8}{Nachricht Rekonstruieren}{43}{Nachricht Rekonstruieren}{0}}} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{1}{43/46}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {43}{46}}} +\HyPL@Entry{46<</P(25)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{2}{47/49}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {47}{49}}} +\HyPL@Entry{49<</P(26)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{3}{50/51}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {50}{51}}} +\HyPL@Entry{51<</P(27)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{4}{52/52}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {52}{52}}} +\HyPL@Entry{52<</P(28)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{5}{53/53}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {53}{53}}} +\HyPL@Entry{53<</P(29)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{6}{54/54}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {54}{54}}} +\@writefile{nav}{\headcommand {\beamer@partpages {1}{54}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {43}{54}}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {43}{54}}} +\@writefile{nav}{\headcommand {\beamer@documentpages {54}}} +\@writefile{nav}{\headcommand {\gdef \inserttotalframenumber {29}}} diff --git a/buch/papers/reedsolomon/RS presentation/RS.log b/buch/papers/reedsolomon/RS presentation/RS.log new file mode 100644 index 0000000..6042adc --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.log @@ -0,0 +1,1252 @@ +This is XeTeX, Version 3.14159265-2.6-0.999991 (TeX Live 2019/W32TeX) (preloaded format=xelatex 2019.10.25) 23 APR 2021 10:48 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**RS.tex +(./RS.tex +LaTeX2e <2019-10-01> patch level 1 +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamer.cls +Document Class: beamer 2019/09/29 v3.57 A class for typesetting presentations +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasemodes.sty +(c:/texlive/2019/texmf-dist/tex/latex/etoolbox/etoolbox.sty +Package: etoolbox 2019/09/21 v2.5h e-TeX tools for LaTeX (JAW) +\etb@tempcnta=\count80 +) +\beamer@tempbox=\box27 +\beamer@tempcount=\count81 +\c@beamerpauses=\count82 + +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasedecode.sty +\beamer@slideinframe=\count83 +\beamer@minimum=\count84 +\beamer@decode@box=\box28 +) +\beamer@commentbox=\box29 +\beamer@modecount=\count85 +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch +) +\headdp=\dimen102 +\footheight=\dimen103 +\sidebarheight=\dimen104 +\beamer@tempdim=\dimen105 +\beamer@finalheight=\dimen106 +\beamer@animht=\dimen107 +\beamer@animdp=\dimen108 +\beamer@animwd=\dimen109 +\beamer@leftmargin=\dimen110 +\beamer@rightmargin=\dimen111 +\beamer@leftsidebar=\dimen112 +\beamer@rightsidebar=\dimen113 +\beamer@boxsize=\dimen114 +\beamer@vboxoffset=\dimen115 +\beamer@descdefault=\dimen116 +\beamer@descriptionwidth=\dimen117 +\beamer@lastskip=\skip41 +\beamer@areabox=\box30 +\beamer@animcurrent=\box31 +\beamer@animshowbox=\box32 +\beamer@sectionbox=\box33 +\beamer@logobox=\box34 +\beamer@linebox=\box35 +\beamer@sectioncount=\count86 +\beamer@subsubsectionmax=\count87 +\beamer@subsectionmax=\count88 +\beamer@sectionmax=\count89 +\beamer@totalheads=\count90 +\beamer@headcounter=\count91 +\beamer@partstartpage=\count92 +\beamer@sectionstartpage=\count93 +\beamer@subsectionstartpage=\count94 +\beamer@animationtempa=\count95 +\beamer@animationtempb=\count96 +\beamer@xpos=\count97 +\beamer@ypos=\count98 +\beamer@ypos@offset=\count99 +\beamer@showpartnumber=\count100 +\beamer@currentsubsection=\count101 +\beamer@coveringdepth=\count102 +\beamer@sectionadjust=\count103 +\beamer@tocsectionnumber=\count104 + +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseoptions.sty +(c:/texlive/2019/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 2014/10/28 v1.15 key=value parser (DPC) +\KV@toks@=\toks14 +)) +\beamer@paperwidth=\skip42 +\beamer@paperheight=\skip43 + +(c:/texlive/2019/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2018/04/16 v5.8 Page Geometry + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +) +(c:/texlive/2019/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +\Gm@cnth=\count105 +\Gm@cntv=\count106 +\c@Gm@tempcnt=\count107 +\Gm@bindingoffset=\dimen118 +\Gm@wd@mp=\dimen119 +\Gm@odd@mp=\dimen120 +\Gm@even@mp=\dimen121 +\Gm@layoutwidth=\dimen122 +\Gm@layoutheight=\dimen123 +\Gm@layouthoffset=\dimen124 +\Gm@layoutvoffset=\dimen125 +\Gm@dimlist=\toks15 +) +(c:/texlive/2019/texmf-dist/tex/latex/base/size11.clo +File: size11.clo 2019/08/27 v1.4j Standard LaTeX file (size option) +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2019/10/08 v1.3c Standard LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 2016/01/03 v1.10 sin cos tan (DPC) +) +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration +) +Package graphics Info: Driver file: xetex.def on input line 105. + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-def/xetex.def +File: xetex.def 2017/06/24 v5.0h Graphics/color driver for xetex +)) +\Gin@req@height=\dimen126 +\Gin@req@width=\dimen127 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks16 +\pgfutil@tempdima=\dimen128 +\pgfutil@tempdimb=\dimen129 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box36 + +(c:/texlive/2019/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/pgf.revision.tex) +Package: pgfrcs 2019/08/03 v3.1.4b (3.1.4b) +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks17 +\pgfkeys@temptoks=\toks18 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks19 +)) +\pgf@x=\dimen130 +\pgf@y=\dimen131 +\pgf@xa=\dimen132 +\pgf@ya=\dimen133 +\pgf@xb=\dimen134 +\pgf@yb=\dimen135 +\pgf@xc=\dimen136 +\pgf@yc=\dimen137 +\pgf@xd=\dimen138 +\pgf@yd=\dimen139 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count108 +\c@pgf@countb=\count109 +\c@pgf@countc=\count110 +\c@pgf@countd=\count111 +\t@pgf@toka=\toks20 +\t@pgf@tokb=\toks21 +\t@pgf@tokc=\toks22 +\pgf@sys@id@count=\count112 + (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2019/08/03 v3.1.4b (3.1.4b) +) +Driver file for pgf: pgfsys-xetex.def + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-xetex.def +File: pgfsys-xetex.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-dvipdfmx.def +File: pgfsys-dvipdfmx.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2019/08/03 v3.1.4b (3.1.4b) +) +\pgfsys@objnum=\count113 +))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfsyssoftpath@smallbuffer@items=\count114 +\pgfsyssoftpath@bigbuffer@items=\count115 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/xcolor/xcolor.sty +Package: xcolor 2016/05/11 v2.12 LaTeX color extensions (UK) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/color.cfg +File: color.cfg 2016/01/02 v1.6 sample color configuration +) +Package xcolor Info: Driver file: xetex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1348. +Package xcolor Info: Model `RGB' extended on input line 1364. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1366. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1367. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1368. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1369. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1370. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1371. +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen140 +\pgfmath@count=\count116 +\pgfmath@box=\box37 +\pgfmath@toks=\toks23 +\pgfmath@stack@operand=\toks24 +\pgfmath@stack@operation=\toks25 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric +.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.t +ex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.co +de.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithm +etics.code.tex))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count117 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@picminx=\dimen141 +\pgf@picmaxx=\dimen142 +\pgf@picminy=\dimen143 +\pgf@picmaxy=\dimen144 +\pgf@pathminx=\dimen145 +\pgf@pathmaxx=\dimen146 +\pgf@pathminy=\dimen147 +\pgf@pathmaxy=\dimen148 +\pgf@xx=\dimen149 +\pgf@xy=\dimen150 +\pgf@yx=\dimen151 +\pgf@yy=\dimen152 +\pgf@zx=\dimen153 +\pgf@zy=\dimen154 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.cod +e.tex +File: pgfcorepathconstruct.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@path@lastx=\dimen155 +\pgf@path@lasty=\dimen156 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.te +x +File: pgfcorepathusage.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@shorten@end@additional=\dimen157 +\pgf@shorten@start@additional=\dimen158 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfpic=\box38 +\pgf@hbox=\box39 +\pgf@layerbox@main=\box40 +\pgf@picture@serial@count=\count118 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code +.tex +File: pgfcoregraphicstate.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgflinewidth=\dimen159 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.c +ode.tex +File: pgfcoretransformations.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@pt@x=\dimen160 +\pgf@pt@y=\dimen161 +\pgf@pt@temp=\dimen162 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.co +de.tex +File: pgfcorepathprocessing.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfarrowsep=\dimen163 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@max=\dimen164 +\pgf@sys@shading@range@num=\count119 +\pgf@shadingcount=\count120 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfexternal@startupbox=\box41 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code +.tex +File: pgfcoretransparency.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +File: pgfcorerdf.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/xxcolor.sty +Package: xxcolor 2003/10/24 ver 0.1 +\XC@nummixins=\count121 +\XC@countmixins=\count122 +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/atbegshi.sty +Package: atbegshi 2016/06/09 v1.18 At begin shipout hook (HO) + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) +)) +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/hyperref.sty +Package: hyperref 2019/09/28 v7.00a Hypertext links for LaTeX + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty +Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) + + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty +Package: hobsub-generic 2016/05/16 v1.14 Bundle oberdiek, subset generic (HO) +Package: hobsub 2016/05/16 v1.14 Construct package bundles (HO) +Package hobsub Info: Skipping package `infwarerr' (already loaded). +Package hobsub Info: Skipping package `ltxcmds' (already loaded). +Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +Package hobsub Info: Skipping package `ifvtex' (already loaded). +Package: intcalc 2016/05/16 v1.2 Expandable calculations with integers (HO) +Package hobsub Info: Skipping package `ifpdf' (already loaded). +Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) +Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) +Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) +Package: pdftexcmds 2019/07/25 v0.30 Utility functions of pdfTeX for LuaTeX (HO +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: pdfTeX >= 1.30 not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode not found. +Package: pdfescape 2016/05/16 v1.14 Implements pdfTeX's escape features (HO) +Package: bigintcalc 2016/05/16 v1.4 Expandable calculations on big integers (HO +) +Package: bitset 2016/05/16 v1.2 Handle bit-vector datatype (HO) +Package: uniquecounter 2016/05/16 v1.3 Provide unlimited unique counter (HO) +) +Package hobsub Info: Skipping package `hobsub' (already loaded). +Package: letltxmacro 2016/05/16 v1.5 Let assignment for LaTeX macros (HO) +Package: hopatch 2016/05/16 v1.3 Wrapper for package hooks (HO) +Package: xcolor-patch 2016/05/16 xcolor patch +Package: atveryend 2016/05/16 v1.9 Hooks at the very end of document (HO) +Package hobsub Info: Skipping package `atbegshi' (already loaded). +Package: refcount 2016/05/16 v3.5 Data extraction from label references (HO) +Package: hycolor 2016/05/16 v1.8 Color options for hyperref/bookmark (HO) +) +(c:/texlive/2019/texmf-dist/tex/latex/oberdiek/auxhook.sty +Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) +) +(c:/texlive/2019/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) +) +\@linkdim=\dimen165 +\Hy@linkcounter=\count123 +\Hy@pagecounter=\count124 + +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/pd1enc.def +File: pd1enc.def 2019/09/28 v7.00a Hyperref: PDFDocEncoding definition (HO) +) +\Hy@SavedSpaceFactor=\count125 + +(c:/texlive/2019/texmf-dist/tex/latex/latexconfig/hyperref.cfg +File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive +) +Package hyperref Info: Option `bookmarks' set `true' on input line 4414. +Package hyperref Info: Option `bookmarksopen' set `true' on input line 4414. +Package hyperref Info: Option `implicit' set `false' on input line 4414. +Package hyperref Info: Hyper figures OFF on input line 4540. +Package hyperref Info: Link nesting OFF on input line 4545. +Package hyperref Info: Hyper index ON on input line 4548. +Package hyperref Info: Plain pages OFF on input line 4555. +Package hyperref Info: Backreferencing OFF on input line 4560. +Package hyperref Info: Implicit mode OFF; no redefinition of LaTeX internals. +Package hyperref Info: Bookmarks ON on input line 4793. +\c@Hy@tempcnt=\count126 + +(c:/texlive/2019/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip10 +Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. +) +LaTeX Info: Redefining \url on input line 5152. +\XeTeXLinkMargin=\dimen166 +\Fld@menulength=\count127 +\Field@Width=\dimen167 +\Fld@charsize=\dimen168 +Package hyperref Info: Hyper figures OFF on input line 6423. +Package hyperref Info: Link nesting OFF on input line 6428. +Package hyperref Info: Hyper index ON on input line 6431. +Package hyperref Info: backreferencing OFF on input line 6438. +Package hyperref Info: Link coloring OFF on input line 6443. +Package hyperref Info: Link coloring with OCG OFF on input line 6448. +Package hyperref Info: PDF/A mode OFF on input line 6453. +LaTeX Info: Redefining \ref on input line 6493. +LaTeX Info: Redefining \pageref on input line 6497. +\Hy@abspage=\count128 + + +Package hyperref Message: Stopped early. + +) +Package hyperref Info: Driver (autodetected): hxetex. + (c:/texlive/2019/texmf-dist/tex/latex/hyperref/hxetex.def +File: hxetex.def 2019/09/28 v7.00a Hyperref driver for XeTeX + +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/puenc.def +File: puenc.def 2019/09/28 v7.00a Hyperref: PDF Unicode definition (HO) +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/stringenc.sty +Package: stringenc 2016/05/16 v1.11 Convert strings between diff. encodings (HO +) +) +\pdfm@box=\box42 +\c@Hy@AnnotLevel=\count129 +\HyField@AnnotCount=\count130 +\Fld@listcount=\count131 +\c@bookmark@seq@number=\count132 + +(c:/texlive/2019/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty +Package: rerunfilecheck 2016/05/16 v1.8 Rerun checks for auxiliary files (HO) +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 +82. +)) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/se-ascii-print.def +File: se-ascii-print.def 2016/05/16 v1.11 stringenc: Printable ASCII characters + +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaserequires.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasecompatibility.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasefont.sty +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2013/01/14 v3.01 AMS font symbols + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\@emptytoks=\toks26 +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +)) +(c:/texlive/2019/texmf-dist/tex/latex/sansmathaccent/sansmathaccent.sty +Package: sansmathaccent 2013/03/28 + +(c:/texlive/2019/texmf-dist/tex/latex/filehook/filehook.sty +Package: filehook 2019/10/03 v0.6 Hooks for input files +))) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetranslator.sty +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator.sty +Package: translator 2019-05-31 v1.12a Easy translation of strings in LaTeX +)) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasemisc.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetwoscreens.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseoverlay.sty +\beamer@argscount=\count133 +\beamer@lastskipcover=\skip44 +\beamer@trivlistdepth=\count134 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetitle.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasesection.sty +\c@lecture=\count135 +\c@part=\count136 +\c@section=\count137 +\c@subsection=\count138 +\c@subsubsection=\count139 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseframe.sty +\beamer@framebox=\box43 +\beamer@frametitlebox=\box44 +\beamer@zoombox=\box45 +\beamer@zoomcount=\count140 +\beamer@zoomframecount=\count141 +\beamer@frametextheight=\dimen169 +\c@subsectionslide=\count142 +\beamer@frametopskip=\skip45 +\beamer@framebottomskip=\skip46 +\beamer@frametopskipautobreak=\skip47 +\beamer@framebottomskipautobreak=\skip48 +\beamer@envbody=\toks27 +\framewidth=\dimen170 +\c@framenumber=\count143 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseverbatim.sty +\beamer@verbatimfileout=\write4 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseframesize.sty +\beamer@splitbox=\box46 +\beamer@autobreakcount=\count144 +\beamer@autobreaklastheight=\dimen171 +\beamer@frametitletoks=\toks28 +\beamer@framesubtitletoks=\toks29 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseframecomponents.sty +\beamer@footins=\box47 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasecolor.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasenotes.sty +\beamer@frameboxcopy=\box48 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetoc.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetemplates.sty +\beamer@sbttoks=\toks30 + +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseauxtemplates.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseboxes.sty +\bmb@box=\box49 +\bmb@colorbox=\box50 +\bmb@boxshadow=\box51 +\bmb@boxshadowball=\box52 +\bmb@boxshadowballlarge=\box53 +\bmb@temp=\dimen172 +\bmb@dima=\dimen173 +\bmb@dimb=\dimen174 +\bmb@prevheight=\dimen175 +) +\beamer@blockheadheight=\dimen176 +)) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaselocalstructure.sty +(c:/texlive/2019/texmf-dist/tex/latex/tools/enumerate.sty +Package: enumerate 2015/07/23 v3.00 enumerate extensions (DPC) +\@enLab=\toks31 +) +\c@figure=\count145 +\c@table=\count146 +\abovecaptionskip=\skip49 +\belowcaptionskip=\skip50 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasenavigation.sty +\beamer@section@min@dim=\dimen177 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetheorems.sty +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2019/04/01 v2.17c AMS math features +\@mathmargin=\skip51 + +For additional information on amsmath, use the `?' option. +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 AMS text + +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks32 +\ex@=\dimen178 +)) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen179 +) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 2016/03/08 v2.02 operator names +) +\inf@bad=\count147 +LaTeX Info: Redefining \frac on input line 227. +\uproot@=\count148 +\leftroot@=\count149 +LaTeX Info: Redefining \overline on input line 389. +\classnum@=\count150 +\DOTSCASE@=\count151 +LaTeX Info: Redefining \ldots on input line 486. +LaTeX Info: Redefining \dots on input line 489. +LaTeX Info: Redefining \cdots on input line 610. +\Mathstrutbox@=\box54 +\strutbox@=\box55 +\big@size=\dimen180 +LaTeX Font Info: Redeclaring font encoding OML on input line 733. +LaTeX Font Info: Redeclaring font encoding OMS on input line 734. +\macc@depth=\count152 +\c@MaxMatrixCols=\count153 +\dotsspace@=\muskip11 +\c@parentequation=\count154 +\dspbrk@lvl=\count155 +\tag@help=\toks33 +\row@=\count156 +\column@=\count157 +\maxfields@=\count158 +\andhelp@=\toks34 +\eqnshift@=\dimen181 +\alignsep@=\dimen182 +\tagshift@=\dimen183 +\tagwidth@=\dimen184 +\totwidth@=\dimen185 +\lineht@=\dimen186 +\@envbody=\toks35 +\multlinegap=\skip52 +\multlinetaggap=\skip53 +\mathdisplay@stack=\toks36 +LaTeX Info: Redefining \[ on input line 2855. +LaTeX Info: Redefining \] on input line 2856. +) +(c:/texlive/2019/texmf-dist/tex/latex/amscls/amsthm.sty +Package: amsthm 2017/10/31 v2.20.4 +\thm@style=\toks37 +\thm@bodyfont=\toks38 +\thm@headfont=\toks39 +\thm@notefont=\toks40 +\thm@headpunct=\toks41 +\thm@preskip=\skip54 +\thm@postskip=\skip55 +\thm@headsep=\skip56 +\dth@everypar=\toks42 +) +\c@theorem=\count159 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasethemes.sty)) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerthemedefault.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerfontthemedefault.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamercolorthemedefault.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerinnerthemedefault.sty +\beamer@dima=\dimen187 +\beamer@dimb=\dimen188 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerouterthemedefault.sty))) +(c:/texlive/2019/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2018/08/11 v1.3c Input encoding file +\inpenc@prehook=\toks43 +\inpenc@posthook=\toks44 + + +Package inputenc Warning: inputenc package ignored with utf8 based engines. + +) (c:/texlive/2019/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2018/08/11 v2.0j Standard LaTeX package + +(c:/texlive/2019/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2018/08/11 v2.0j Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 48. +) +LaTeX Font Info: Trying to load font information for T1+lmss on input line 1 +05. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/t1lmss.fd +File: t1lmss.fd 2009/10/30 v1.6 Font defs for Latin Modern +)) +(c:/texlive/2019/texmf-dist/tex/latex/lm/lmodern.sty +Package: lmodern 2009/10/30 v1.6 Latin Modern Fonts +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/cmr/m/n --> OT1/lmr/m/n on input line 22. +LaTeX Font Info: Overwriting symbol font `letters' in version `normal' +(Font) OML/cmm/m/it --> OML/lmm/m/it on input line 23. +LaTeX Font Info: Overwriting symbol font `symbols' in version `normal' +(Font) OMS/cmsy/m/n --> OMS/lmsy/m/n on input line 24. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `normal' +(Font) OMX/cmex/m/n --> OMX/lmex/m/n on input line 25. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/lmr/bx/n on input line 26. +LaTeX Font Info: Overwriting symbol font `letters' in version `bold' +(Font) OML/cmm/b/it --> OML/lmm/b/it on input line 27. +LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' +(Font) OMS/cmsy/b/n --> OMS/lmsy/b/n on input line 28. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' +(Font) OMX/cmex/m/n --> OMX/lmex/m/n on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/cmr/bx/n --> OT1/lmr/bx/n on input line 31. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' +(Font) OT1/cmss/m/n --> OT1/lmss/m/n on input line 32. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' +(Font) OT1/cmr/m/it --> OT1/lmr/m/it on input line 33. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' +(Font) OT1/cmtt/m/n --> OT1/lmtt/m/n on input line 34. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/lmr/bx/n on input line 35. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/cmss/bx/n --> OT1/lmss/bx/n on input line 36. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/cmr/bx/it --> OT1/lmr/bx/it on input line 37. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/cmtt/m/n --> OT1/lmtt/m/n on input line 38. +) +(c:/texlive/2019/texmf-dist/tex/generic/babel/babel.sty +Package: babel 2019/10/15 3.35 The Babel package + +(c:/texlive/2019/texmf-dist/tex/generic/babel/switch.def +File: switch.def 2019/10/15 3.35 Babel switching mechanism +) +(c:/texlive/2019/texmf-dist/tex/generic/babel-german/ngerman.ldf +Language: ngerman 2018/12/08 v2.11 German support for babel (post-1996 orthogra +phy) + +(c:/texlive/2019/texmf-dist/tex/generic/babel-german/ngermanb.ldf +Language: ngermanb 2018/12/08 v2.11 German support for babel (post-1996 orthogr +aphy) + +(c:/texlive/2019/texmf-dist/tex/generic/babel/babel.def +File: babel.def 2019/10/15 3.35 Babel common definitions +\babel@savecnt=\count160 +\U@D=\dimen189 + +(c:/texlive/2019/texmf-dist/tex/generic/babel/xebabel.def +(c:/texlive/2019/texmf-dist/tex/generic/babel/txtbabel.def)) +LaTeX Info: Redefining \textlatin on input line 2185. +\bbl@dirlevel=\count161 +) +Package babel Info: Making " an active character on input line 121. +))) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +Package: pgf 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +File: pgfmoduleshapes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfnodeparttextbox=\box56 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.st +y +Package: pgfcomp-version-0-65 2019/08/03 v3.1.4b (3.1.4b) +\pgf@nodesepstart=\dimen190 +\pgf@nodesepend=\dimen191 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.st +y +Package: pgfcomp-version-1-18 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/math/pgfmath.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex) +\pgffor@iter=\dimen192 +\pgffor@skip=\dimen193 +\pgffor@stack=\toks45 +\pgffor@toks=\toks46 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.co +de.tex +File: pgflibraryplothandlers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@plot@mark@count=\count162 +\pgfplotmarksize=\dimen194 +) +\tikz@lastx=\dimen195 +\tikz@lasty=\dimen196 +\tikz@lastxsaved=\dimen197 +\tikz@lastysaved=\dimen198 +\tikz@lastmovetox=\dimen199 +\tikz@lastmovetoy=\dimen256 +\tikzleveldistance=\dimen257 +\tikzsiblingdistance=\dimen258 +\tikz@figbox=\box57 +\tikz@figbox@bg=\box58 +\tikz@tempbox=\box59 +\tikz@tempbox@bg=\box60 +\tikztreelevel=\count163 +\tikznumberofchildren=\count164 +\tikznumberofcurrentchild=\count165 +\tikz@fig@count=\count166 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfmatrixcurrentrow=\count167 +\pgfmatrixcurrentcolumn=\count168 +\pgf@matrix@numberofcolumns=\count169 +) +\tikz@expandcount=\count170 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarytopaths.code.tex +File: tikzlibrarytopaths.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerthemeHannover.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerouterthemesidebar.sty +\beamer@sidebarwidth=\dimen259 +\beamer@headheight=\dimen260 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamercolorthemeseahorse.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerinnerthemecircles.sty)) +(./RS.aux) +\openout1 = `RS.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. + +*geometry* driver: auto-detecting +*geometry* detected driver: xetex +*geometry* verbose mode - [ preamble ] result: +* driver: xetex +* paper: custom +* layout: <same size as paper> +* layoutoffset:(h,v)=(0.0pt,0.0pt) +* modes: includehead includefoot +* h-part:(L,W,R)=(59.22636pt, 381.79135pt, 14.22636pt) +* v-part:(T,H,B)=(0.0pt, 256.0748pt, 0.0pt) +* \paperwidth=455.24408pt +* \paperheight=256.0748pt +* \textwidth=381.79135pt +* \textheight=227.62207pt +* \oddsidemargin=-13.04362pt +* \evensidemargin=-13.04362pt +* \topmargin=-72.26999pt +* \headheight=14.22636pt +* \headsep=0.0pt +* \topskip=11.0pt +* \footskip=14.22636pt +* \marginparwidth=4.0pt +* \marginparsep=10.0pt +* \columnsep=10.0pt +* \skip\footins=10.0pt plus 4.0pt minus 2.0pt +* \hoffset=0.0pt +* \voffset=0.0pt +* \mag=1000 +* \@twocolumnfalse +* \@twosidefalse +* \@mparswitchfalse +* \@reversemarginfalse +* (1in=72.27pt=25.4mm, 1cm=28.453pt) + +ABD: EveryShipout initializing macros +\AtBeginShipoutBox=\box61 +Package hyperref Info: Link coloring OFF on input line 9. + +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/nameref.sty +Package: nameref 2019/09/16 v2.46 Cross-referencing by name of section + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/gettitlestring.sty +Package: gettitlestring 2016/05/16 v1.5 Cleanup title references (HO) +) +\c@section@level=\count171 +) +LaTeX Info: Redefining \ref on input line 9. +LaTeX Info: Redefining \pageref on input line 9. +LaTeX Info: Redefining \nameref on input line 9. + (./RS.out) (./RS.out) +\@outlinefile=\write5 +\openout5 = `RS.out'. + +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/lmr/m/n --> OT1/cmss/m/n on input line 9. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/lmr/bx/n --> OT1/cmss/bx/n on input line 9. +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/cmss/m/n --> OT1/lmss/m/n on input line 9. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/cmss/bx/n --> OT1/lmss/bx/n on input line 9. +\symnumbers=\mathgroup6 +\sympureletters=\mathgroup7 +LaTeX Font Info: Overwriting math alphabet `\mathrm' in version `normal' +(Font) OT1/lmss/m/n --> T1/lmr/m/n on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/lmr/bx/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/lmr/bx/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathsf on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' +(Font) OT1/lmss/m/n --> T1/lmss/m/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/lmss/bx/n --> T1/lmss/m/n on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathit on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' +(Font) OT1/lmr/m/it --> T1/lmss/m/it on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/lmr/bx/it --> T1/lmss/m/it on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathtt on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' +(Font) OT1/lmtt/m/n --> T1/lmtt/m/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/lmtt/m/n --> T1/lmtt/m/n on input line 9. +LaTeX Font Info: Overwriting symbol font `numbers' in version `bold' +(Font) T1/lmss/m/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting symbol font `pureletters' in version `bold' +(Font) T1/lmss/m/it --> T1/lmss/bx/it on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathrm' in version `bold' +(Font) OT1/lmss/bx/n --> T1/lmr/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) T1/lmss/bx/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) T1/lmss/m/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) T1/lmss/m/it --> T1/lmss/bx/it on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) T1/lmtt/m/n --> T1/lmtt/bx/n on input line 9. + +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-basic-dictionary-En +glish.dict +Dictionary: translator-basic-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-bibliography-dictio +nary-English.dict +Dictionary: translator-bibliography-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-environment-diction +ary-English.dict +Dictionary: translator-environment-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-months-dictionary-E +nglish.dict +Dictionary: translator-months-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-numbers-dictionary- +English.dict +Dictionary: translator-numbers-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-theorem-dictionary- +English.dict +Dictionary: translator-theorem-dictionary, Language: English +) (./RS.nav) + +Package hyperref Warning: Option `pdfauthor' has already been used, +(hyperref) setting the option has no effect on input line 10. + + +Package hyperref Warning: Option `pdfsubject' has already been used, +(hyperref) setting the option has no effect on input line 16. + +[1 + +] +LaTeX Font Info: Trying to load font information for OT1+lmss on input line +32. + (c:/texlive/2019/texmf-dist/tex/latex/lm/ot1lmss.fd +File: ot1lmss.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: Trying to load font information for OML+lmm on input line 3 +2. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/omllmm.fd +File: omllmm.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: Trying to load font information for OMS+lmsy on input line +32. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/omslmsy.fd +File: omslmsy.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: Trying to load font information for OMX+lmex on input line +32. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/omxlmex.fd +File: omxlmex.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <10.95> on input line 32. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <8> on input line 32. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <6> on input line 32. +LaTeX Font Info: Trying to load font information for U+msa on input line 32. + + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Trying to load font information for U+msb on input line 32. + + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <10.95> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 32. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <8> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 32. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <6> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 32. + [2 + +] [3 + +] [4 + +] [5 + +] +[6 + +] +File: images/polynom1.pdf Graphic file (type pdf) +<use images/polynom1.pdf> + [7 + +] +File: images/polynom2.pdf Graphic file (type pdf) +<use images/polynom2.pdf> + [8 + +] [9 + +] [10 + +] [11 + +] [12 + +] [13 + +] +File: images/fig1.pdf Graphic file (type pdf) +<use images/fig1.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[14 + +] +File: images/fig2.pdf Graphic file (type pdf) +<use images/fig2.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[15 + +] +File: images/fig3.pdf Graphic file (type pdf) +<use images/fig3.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[16 + +] +File: images/fig4.pdf Graphic file (type pdf) +<use images/fig4.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[17 + +] +File: images/fig5.pdf Graphic file (type pdf) +<use images/fig5.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[18 + +] +File: images/fig6.pdf Graphic file (type pdf) +<use images/fig6.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[19 + +] +File: images/fig7.pdf Graphic file (type pdf) +<use images/fig7.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 124 + [] + +[20 + +] [21 + +] [22 + +] [23 + +] [24 + +] [25 + +] [26 + +] [27 + +] [28 + +] +LaTeX Font Info: Trying to load font information for T1+lmr on input line 30 +0. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/t1lmr.fd +File: t1lmr.fd 2009/10/30 v1.6 Font defs for Latin Modern +) [29 + +] [30 + +] +LaTeX Font Info: Trying to load font information for U+euf on input line 383 +. + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/ueuf.fd +File: ueuf.fd 2013/01/14 v3.01 Euler Fraktur +) +Overfull \hbox (1.42268pt too wide) detected at line 383 +\U/euf/m/n/10.95 F[]\OT1/lmss/m/n/10.95 (\T1/lmss/m/sl/10.95 F\OT1/lmss/m/n/10. +95 (\OML/lmm/m/it/10.95 !\OT1/lmss/m/n/10.95 )) = \T1/lmss/m/sl/10.95 f\OT1/lms +s/m/n/10.95 (\T1/lmss/m/sl/10.95 t\OT1/lmss/m/n/10.95 ) = [] [][] \T1/lmss/m/sl +/10.95 F\OT1/lmss/m/n/10.95 (\OML/lmm/m/it/10.95 !\OT1/lmss/m/n/10.95 )\T1/lmr/ +m/n/10.95 e[]\T1/lmss/m/sl/10.95 d\OML/lmm/m/it/10.95 ! + [] + +[31 + +] [32 + +] [33 + +] [34 + +] [35 + +] [36 + +] [37 + +] [38 + +] +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <14.4> on input line 638. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <10> on input line 638. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <7> on input line 638. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <14.4> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 638. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <10> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 638. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <7> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 638. + [39 + +] +Overfull \hbox (2.91844pt too wide) detected at line 674 +[] + [] + +[40 + +] [41 + +] [42 + +] [43 + +] [44 + +] [45 + +] [46 + +] [47 + +] [48 + +] [49 + +] [50 + +] [51 + +] [52 + +] [53 + +] [54 + +] +\tf@nav=\write6 +\openout6 = `RS.nav'. + +\tf@toc=\write7 +\openout7 = `RS.toc'. + +\tf@snm=\write8 +\openout8 = `RS.snm'. + +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 928. +Package atveryend Info: Empty hook `AfterLastShipout' on input line 928. + +(./RS.aux) +Package atveryend Info: Empty hook `AtVeryEndDocument' on input line 928. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 928. +Package rerunfilecheck Info: File `RS.out' has not changed. +(rerunfilecheck) Checksum: CBEDF1F633104E8EE4EB074E401487DA. + ) +Here is how much of TeX's memory you used: + 24528 strings out of 492483 + 453801 string characters out of 6132858 + 557615 words of memory out of 5000000 + 28488 multiletter control sequences out of 15000+600000 + 82614 words of font info for 73 fonts, out of 8000000 for 9000 + 1348 hyphenation exceptions out of 8191 + 58i,15n,61p,796b,566s stack positions out of 5000i,500n,10000p,200000b,80000s + +Output written on RS.pdf (54 pages). diff --git a/buch/papers/reedsolomon/RS presentation/RS.nav b/buch/papers/reedsolomon/RS presentation/RS.nav new file mode 100644 index 0000000..f288963 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.nav @@ -0,0 +1,87 @@ +\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}} +\headcommand {\beamer@framepages {1}{1}} +\headcommand {\beamer@sectionpages {1}{1}} +\headcommand {\beamer@subsectionpages {1}{1}} +\headcommand {\sectionentry {1}{Einführung}{2}{Einführung}{0}} +\headcommand {\slideentry {1}{0}{1}{2/4}{}{0}} +\headcommand {\beamer@framepages {2}{4}} +\headcommand {\beamer@sectionpages {2}{4}} +\headcommand {\beamer@subsectionpages {2}{4}} +\headcommand {\sectionentry {2}{Polynom Ansatz}{5}{Polynom Ansatz}{0}} +\headcommand {\slideentry {2}{0}{1}{5/5}{}{0}} +\headcommand {\beamer@framepages {5}{5}} +\headcommand {\slideentry {2}{0}{2}{6/8}{}{0}} +\headcommand {\beamer@framepages {6}{8}} +\headcommand {\slideentry {2}{0}{3}{9/12}{}{0}} +\headcommand {\beamer@framepages {9}{12}} +\headcommand {\beamer@sectionpages {5}{12}} +\headcommand {\beamer@subsectionpages {5}{12}} +\headcommand {\sectionentry {3}{Diskrete Fourier Transformation}{13}{Diskrete Fourier Transformation}{0}} +\headcommand {\slideentry {3}{0}{1}{13/13}{}{0}} +\headcommand {\beamer@framepages {13}{13}} +\headcommand {\slideentry {3}{0}{2}{14/20}{}{0}} +\headcommand {\beamer@framepages {14}{20}} +\headcommand {\slideentry {3}{0}{3}{21/23}{}{0}} +\headcommand {\beamer@framepages {21}{23}} +\headcommand {\slideentry {3}{0}{4}{24/24}{}{0}} +\headcommand {\beamer@framepages {24}{24}} +\headcommand {\slideentry {3}{0}{5}{25/26}{}{0}} +\headcommand {\beamer@framepages {25}{26}} +\headcommand {\beamer@sectionpages {13}{26}} +\headcommand {\beamer@subsectionpages {13}{26}} +\headcommand {\sectionentry {4}{Reed-Solomon in Endlichen Körpern}{27}{Reed-Solomon in Endlichen Körpern}{0}} +\headcommand {\slideentry {4}{0}{1}{27/27}{}{0}} +\headcommand {\beamer@framepages {27}{27}} +\headcommand {\slideentry {4}{0}{2}{28/28}{}{0}} +\headcommand {\beamer@framepages {28}{28}} +\headcommand {\beamer@sectionpages {27}{28}} +\headcommand {\beamer@subsectionpages {27}{28}} +\headcommand {\sectionentry {5}{Codierung eines Beispiels}{29}{Codierung eines Beispiels}{0}} +\headcommand {\slideentry {5}{0}{1}{29/29}{}{0}} +\headcommand {\beamer@framepages {29}{29}} +\headcommand {\slideentry {5}{0}{2}{30/30}{}{0}} +\headcommand {\beamer@framepages {30}{30}} +\headcommand {\beamer@sectionpages {29}{30}} +\headcommand {\beamer@subsectionpages {29}{30}} +\headcommand {\sectionentry {6}{Decodierung ohne Fehler}{31}{Decodierung ohne Fehler}{0}} +\headcommand {\slideentry {6}{0}{1}{31/31}{}{0}} +\headcommand {\beamer@framepages {31}{31}} +\headcommand {\slideentry {6}{0}{2}{32/34}{}{0}} +\headcommand {\beamer@framepages {32}{34}} +\headcommand {\slideentry {6}{0}{3}{35/35}{}{0}} +\headcommand {\beamer@framepages {35}{35}} +\headcommand {\beamer@sectionpages {31}{35}} +\headcommand {\beamer@subsectionpages {31}{35}} +\headcommand {\sectionentry {7}{Decodierung mit Fehler}{36}{Decodierung mit Fehler}{0}} +\headcommand {\slideentry {7}{0}{1}{36/36}{}{0}} +\headcommand {\beamer@framepages {36}{36}} +\headcommand {\slideentry {7}{0}{2}{37/37}{}{0}} +\headcommand {\beamer@framepages {37}{37}} +\headcommand {\slideentry {7}{0}{3}{38/38}{}{0}} +\headcommand {\beamer@framepages {38}{38}} +\headcommand {\slideentry {7}{0}{4}{39/39}{}{0}} +\headcommand {\beamer@framepages {39}{39}} +\headcommand {\slideentry {7}{0}{5}{40/40}{}{0}} +\headcommand {\beamer@framepages {40}{40}} +\headcommand {\slideentry {7}{0}{6}{41/42}{}{0}} +\headcommand {\beamer@framepages {41}{42}} +\headcommand {\beamer@sectionpages {36}{42}} +\headcommand {\beamer@subsectionpages {36}{42}} +\headcommand {\sectionentry {8}{Nachricht Rekonstruieren}{43}{Nachricht Rekonstruieren}{0}} +\headcommand {\slideentry {8}{0}{1}{43/46}{}{0}} +\headcommand {\beamer@framepages {43}{46}} +\headcommand {\slideentry {8}{0}{2}{47/49}{}{0}} +\headcommand {\beamer@framepages {47}{49}} +\headcommand {\slideentry {8}{0}{3}{50/51}{}{0}} +\headcommand {\beamer@framepages {50}{51}} +\headcommand {\slideentry {8}{0}{4}{52/52}{}{0}} +\headcommand {\beamer@framepages {52}{52}} +\headcommand {\slideentry {8}{0}{5}{53/53}{}{0}} +\headcommand {\beamer@framepages {53}{53}} +\headcommand {\slideentry {8}{0}{6}{54/54}{}{0}} +\headcommand {\beamer@framepages {54}{54}} +\headcommand {\beamer@partpages {1}{54}} +\headcommand {\beamer@subsectionpages {43}{54}} +\headcommand {\beamer@sectionpages {43}{54}} +\headcommand {\beamer@documentpages {54}} +\headcommand {\gdef \inserttotalframenumber {29}} diff --git a/buch/papers/reedsolomon/RS presentation/RS.out b/buch/papers/reedsolomon/RS presentation/RS.out new file mode 100644 index 0000000..364319e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.out @@ -0,0 +1,8 @@ +\BOOKMARK [2][]{Outline0.1}{Einführung}{}% 1 +\BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2 +\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5 +\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6 +\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7 +\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8 diff --git a/buch/papers/reedsolomon/RS presentation/RS.pdf b/buch/papers/reedsolomon/RS presentation/RS.pdf Binary files differnew file mode 100644 index 0000000..d9d6693 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.pdf diff --git a/buch/papers/reedsolomon/RS presentation/RS.snm b/buch/papers/reedsolomon/RS presentation/RS.snm new file mode 100644 index 0000000..86859c9 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.snm @@ -0,0 +1 @@ +\beamer@slide {ft_discrete}{21} diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz Binary files differnew file mode 100644 index 0000000..04bd239 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex new file mode 100644 index 0000000..6ee6cc7 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -0,0 +1,934 @@ +\documentclass[11pt,aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[ngerman]{babel} +\usepackage{tikz} +\usetheme{Hannover} + +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Reed-Solomon-Code:} + \begin{itemize} + \visible<1->{\item Für Übertragung von Daten} + \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} + \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- +\section{Polynom Ansatz} + \begin{frame} + \begin{itemize} + \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} + \end{frame} + \begin{frame} + \frametitle{Beispiel} + Übertragen von + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + + + Versende $ (p(1),p(2),\dots,p(7))$ + \visible<2->{ = (\textcolor{green}{8},} + \only<2>{\textcolor{green}{15},} + \only<3>{\textcolor{red}{50},} + \only<2>{\textcolor{green}{26},} + \only<3>{\textcolor{red}{37},} + \visible<2->{\textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})} + \only<2>{\includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<3>{\includegraphics[scale = 1.2]{images/polynom2.pdf}} + \visible<3>{ + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ +\visible<1->{3}& +\visible<1->{3}& +\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ + &&\\ +\visible<1->{$k$} & +\visible<1->{$t$} & +\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ + \hline + &&\\ + &&\\ + \multicolumn{3}{l} { + \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + } + \end{tabular} + \end{center} + \end{frame} + +%------------------------------------------------------------------------------- + +\section{Diskrete Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + \visible<1->{ + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]} + \visible<2->{ + \item Ersetzte + \[ + w = e^{-\frac{2\pi j}{N} k} + \]} + \visible<3->{ + \item Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \]} + \end{itemize} + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = \frac{1}{N} + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] + \end{frame} +%------------------------------------------------------------------------------- + + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \visible<2>{ + \newline + Indem in einem endlichen Körper gerechnet wird. + } + \end{frame} + +%------------------------------------------------------------------------------- + + +\section{Reed-Solomon in Endlichen Körpern} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \onslide<1->{\item Warum endliche Körper?} + + \onslide<2->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} + + \onslide<3->{\qquad digitale Fehlerkorrektur} + + %\onslide<4->{\qquad bessere Laufzeit} + + \vspace{10pt} + + \onslide<4->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} + + \vspace{10pt} + + \onslide<5->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} + + \onslide<6->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \onslide<1->{\item endlicher Körper $q = 11$} + + \onslide<2->{ist eine Primzahl} + + \onslide<3->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} + + \vspace{10pt} + + \onslide<4->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen} + + \onslide<5->{$n = q - 1 = 10$ Zahlen} + + \vspace{10pt} + + \onslide<6->{\item Max.~Fehler $t = 2$} + + \onslide<7->{maximale Anzahl von Fehler, die wir noch korrigieren können} + + \vspace{10pt} + + \onslide<8->{\item Nutzlast $k = n -2t = 6$ Zahlen} + + \onslide<9->{Fehlerkorrkturstellen $2t = 4$ Zahlen} + + \onslide<10->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} + + \onslide<11->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \onslide<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} + + \vspace{10pt} + + \onslide<2->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} + + \vspace{10pt} + + \onslide<3->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken} + + \onslide<4->{$\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} + + \vspace{10pt} + + \onslide<5->{\item Wir wählen $a = 8$} + + \onslide<6->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} + + \onslide<7->{$8$ ist eine primitive Einheitswurzel} + + \vspace{10pt} + + \onslide<8->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} + + \onslide<9->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \onslide<1->{\item Übertragungsvektor $v$} + + \onslide<2->{\item $v = A \cdot m$} + + \end{itemize} + + \[ + \onslide<3->{ + v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + } + \] + + \begin{itemize} + \onslide<4->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \onslide<1->{\item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$} + + \vspace{10pt} + + \onslide<2->{\item Wir suchen die Inverse der Matrix $A$} + + \vspace{10pt} + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.55\textwidth} + \onslide<3->{ Inverse der Fouriertransformation} + \vspace{10pt} + \onslide<4->{ + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + } + \vspace{10pt} + \onslide<5->{ + \[ + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + } + \end{column} + \begin{column}{0.45\textwidth} + \onslide<6->{Inverse von $a$} + + \vspace{10pt} + + \onslide<7->{ + \[ + 8^{1} \Rightarrow 8^{-1} + \] + } + + \onslide<8->{Inverse finden wir über den Eulkidischen Algorithmus} + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + \onslide<1->{ + \begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor<2->{blue}{$-4$}& \textcolor<2->{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + } + + \vspace{10pt} + + \begin{tabular}{rcl} + \onslide<3->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ + \onslide<4->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ + \onslide<5->{$8^{-1}$ &$=$& $7$} + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Inverser Matrix} + + \begin{itemize} + \onslide<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} + + \onslide<2->{\item $m = 1/10 \cdot A^{-1} \cdot v$} + + \onslide<3->{\item $m = 10 \cdot A^{-1} \cdot v$} + + \end{itemize} + \onslide<4->{ + \[ + m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + } + + \begin{itemize} + \onslide<5->{\item $m = [0,0,0,0,4,7,2,5,8,1]$} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \onslide<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} + + \onslide<2->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + + \onslide<3->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} + + \end{itemize} + + \onslide<4->{Wie finden wir die Fehler?} + + \begin{itemize} + \onslide<5->{\item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + + \onslide<6->{\item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$} + + %\only<7->{\item $e(X) = r(X) - m(X)$} + + \onslide<7->{\item $e(X) = r(X) - m(X)$} + + \end{itemize} + + \begin{center} + \onslide<8->{ + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& \onslide<9->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ + $m(a^{i})$& \onslide<10->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ + $e(a^{i})$& \onslide<11->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ + \hline + \end{tabular} + } + \end{center} + + + \begin{itemize} + \onslide<12->{\item Alle Stellen, die nicht Null sind, sind Fehler} + \end{itemize} + + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} + + \vspace{10pt} + + \onslide<2->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} + + \vspace{10pt} + + \onslide<3->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} + + \vspace{10pt} + + \onslide<4->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} + + \vspace{10pt} + + \onslide<5->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} + + \vspace{10pt} + + \onslide<6->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} + + \vspace{10pt} + + \onslide<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$} + + \vspace{10pt} + + \onslide<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} + + \vspace{10pt} + + \onslide<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} + + \vspace{10pt} + + \onslide<1->{\item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können} + + \vspace{10pt} + + \onslide<2->{$\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$} + + \onslide<3->{$= d(X) \cdot e(X)$} + + \vspace{10pt} + + \onslide<4->{\item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Kennen wir $e(X)$?} + + \begin{itemize} + + \onslide<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} + + \vspace{10pt} + + \onslide<2->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} + + \vspace{10pt} + + \onslide<3->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat} + + \vspace{10pt} + + \onslide<4->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} + + \vspace{10pt} + + \onslide<5->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} + + \vspace{10pt} + + \onslide<6->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + \onslide<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} + \onslide<2->{ + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + } + \onslide<3->{ + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + } + \vspace{10pt} + + \onslide<4->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} + + \vspace{10pt} + + \onslide<5->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor<2->{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + \onslide<3->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} + \onslide<4->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} + \onslide<5->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} + \end{tabular} + + \vspace{10pt} + + \onslide<6->{ + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + } + + \onslide<7->{$d(X) = (X-a^3)(X-a^8)$} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \onslide<1->{\item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + + \onslide<2->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} + + \end{itemize} + \onslide<3->{ + \[ + \textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor<4->{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor<4->{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor<4->{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor<4->{red}{1} \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^3}& \textcolor<4->{red}{8^6}& \textcolor<4->{red}{8^9}& \textcolor<4->{red}{8^{12}}& \textcolor<4->{red}{8^{15}}& \textcolor<4->{red}{8^{18}}& \textcolor<4->{red}{8^{21}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^8}& \textcolor<4->{red}{8^{16}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{32}}& \textcolor<4->{red}{8^{40}}& \textcolor<4->{red}{8^{48}}& \textcolor<4->{red}{8^{56}}& \textcolor<4->{red}{8^{64}}& \textcolor<4->{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + } + + \begin{itemize} + \onslide<5->{\item Fehlerstellen entfernen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<4->{green}{8^6}& \textcolor<4->{green}{8^7}& \textcolor<4->{green}{8^8}& \textcolor<4->{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<4->{green}{8^{12}}& \textcolor<4->{green}{8^{14}}& \textcolor<4->{green}{8^{16}}& \textcolor<4->{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<4->{green}{8^{24}}& \textcolor<4->{green}{8^{28}}& \textcolor<4->{green}{8^{32}}& \textcolor<4->{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<4->{green}{8^{30}}& \textcolor<4->{green}{8^{35}}& \textcolor<4->{green}{8^{40}}& \textcolor<4->{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<4->{green}{8^{36}}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{48}}& \textcolor<4->{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{49}}& \textcolor<4->{green}{8^{56}}& \textcolor<4->{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<4->{green}{8^{54}}& \textcolor<4->{green}{8^{63}}& \textcolor<4->{green}{8^{72}}& \textcolor<4->{green}{8^{81}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\ + \end{pmatrix} + \] + + \begin{itemize} + \onslide<3->{\item Nullstellen entfernen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<3->{red}{7} \\ \textcolor<3->{red}{4} \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^7}& \textcolor<3->{red}{8^{14}}& \textcolor<3->{red}{8^{21}}& \textcolor<3->{red}{8^{28}}& \textcolor<3->{red}{8^{35}}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^9}& \textcolor<3->{red}{8^{18}}& \textcolor<3->{red}{8^{27}}& \textcolor<3->{red}{8^{36}}& \textcolor<3->{red}{8^{45}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \onslide<2->{\item Matrix in eine Quadratische Form bringen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \onslide<2->{\item Matrix Invertieren} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + \onslide<2->{$\Downarrow$} + \end{center} + \[ + \onslide<3->{ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + } + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \onslide<2->{\item $m = [4,7,2,5,8,1]$} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + +\end{document} diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc new file mode 100644 index 0000000..095b5e6 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -0,0 +1,9 @@ +\babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{5}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{13}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{27}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{29}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{31}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{36}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{43}{0}{8} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.aux b/buch/papers/reedsolomon/RS presentation/RS_handout.aux new file mode 100644 index 0000000..41ccfb5 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.aux @@ -0,0 +1,143 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand*\HyPL@Entry[1]{} +\bbl@beforestart +\catcode `"\active +\HyPL@Entry{0<</P(1)>>} +\babel@aux{ngerman}{} +\@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {1}{1}}} +\HyPL@Entry{1<</P(2)>>} +\@writefile{toc}{\beamer@sectionintoc {1}{Einführung}{2}{0}{1}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {1}{1}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{1}}} +\@writefile{nav}{\headcommand {\sectionentry {1}{Einführung}{2}{Einführung}{0}}} +\@writefile{nav}{\headcommand {\slideentry {1}{0}{1}{2/2}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {2}{2}}} +\HyPL@Entry{2<</P(3)>>} +\@writefile{toc}{\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {2}{2}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {2}{2}}} +\@writefile{nav}{\headcommand {\sectionentry {2}{Polynom Ansatz}{3}{Polynom Ansatz}{0}}} +\@writefile{nav}{\headcommand {\slideentry {2}{0}{1}{3/3}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {3}{3}}} +\HyPL@Entry{3<</P(4)>>} +\@writefile{nav}{\headcommand {\slideentry {2}{0}{2}{4/4}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {4}{4}}} +\HyPL@Entry{4<</P(5)>>} +\@writefile{toc}{\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{5}{0}{3}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {3}{4}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {3}{4}}} +\@writefile{nav}{\headcommand {\sectionentry {3}{Diskrete Fourier Transformation}{5}{Diskrete Fourier Transformation}{0}}} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{1}{5/5}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {5}{5}}} +\HyPL@Entry{5<</P(6)>>} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{2}{6/12}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {6}{12}}} +\HyPL@Entry{12<</P(7)>>} +\@writefile{snm}{\beamer@slide {ft_discrete}{13}} +\newlabel{ft_discrete}{{7}{13}{Diskrete Fourier Transformation}{Doc-Start}{}} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{3}{13/13}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {13}{13}}} +\HyPL@Entry{13<</P(8)>>} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{4}{14/14}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {14}{14}}} +\HyPL@Entry{14<</P(9)>>} +\@writefile{nav}{\headcommand {\slideentry {3}{0}{5}{15/15}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {15}{15}}} +\HyPL@Entry{15<</P(10)>>} +\@writefile{toc}{\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{16}{0}{4}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {5}{15}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {5}{15}}} +\@writefile{nav}{\headcommand {\sectionentry {4}{Reed-Solomon in Endlichen Körpern}{16}{Reed-Solomon in Endlichen Körpern}{0}}} +\@writefile{nav}{\headcommand {\slideentry {4}{0}{1}{16/16}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {16}{16}}} +\HyPL@Entry{16<</P(11)>>} +\@writefile{nav}{\headcommand {\slideentry {4}{0}{2}{17/17}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {17}{17}}} +\HyPL@Entry{17<</P(12)>>} +\@writefile{toc}{\beamer@sectionintoc {5}{Codierung eines Beispiels}{18}{0}{5}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {16}{17}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {16}{17}}} +\@writefile{nav}{\headcommand {\sectionentry {5}{Codierung eines Beispiels}{18}{Codierung eines Beispiels}{0}}} +\@writefile{nav}{\headcommand {\slideentry {5}{0}{1}{18/18}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {18}{18}}} +\HyPL@Entry{18<</P(13)>>} +\@writefile{nav}{\headcommand {\slideentry {5}{0}{2}{19/19}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {19}{19}}} +\HyPL@Entry{19<</P(14)>>} +\@writefile{toc}{\beamer@sectionintoc {6}{Decodierung ohne Fehler}{20}{0}{6}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {18}{19}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {18}{19}}} +\@writefile{nav}{\headcommand {\sectionentry {6}{Decodierung ohne Fehler}{20}{Decodierung ohne Fehler}{0}}} +\@writefile{nav}{\headcommand {\slideentry {6}{0}{1}{20/20}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {20}{20}}} +\HyPL@Entry{20<</P(15)>>} +\@writefile{nav}{\headcommand {\slideentry {6}{0}{2}{21/21}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {21}{21}}} +\HyPL@Entry{21<</P(16)>>} +\@writefile{nav}{\headcommand {\slideentry {6}{0}{3}{22/22}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {22}{22}}} +\HyPL@Entry{22<</P(17)>>} +\@writefile{toc}{\beamer@sectionintoc {7}{Decodierung mit Fehler}{23}{0}{7}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {20}{22}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {20}{22}}} +\@writefile{nav}{\headcommand {\sectionentry {7}{Decodierung mit Fehler}{23}{Decodierung mit Fehler}{0}}} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{1}{23/23}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {23}{23}}} +\HyPL@Entry{23<</P(18)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{2}{24/24}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {24}{24}}} +\HyPL@Entry{24<</P(19)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{3}{25/25}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {25}{25}}} +\HyPL@Entry{25<</P(20)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{4}{26/26}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {26}{26}}} +\HyPL@Entry{26<</P(21)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{5}{27/27}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {27}{27}}} +\HyPL@Entry{27<</P(22)>>} +\@writefile{nav}{\headcommand {\slideentry {7}{0}{6}{28/28}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {28}{28}}} +\HyPL@Entry{28<</P(23)>>} +\@writefile{toc}{\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{29}{0}{8}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {23}{28}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {23}{28}}} +\@writefile{nav}{\headcommand {\sectionentry {8}{Nachricht Rekonstruieren}{29}{Nachricht Rekonstruieren}{0}}} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{1}{29/29}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {29}{29}}} +\HyPL@Entry{29<</P(24)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{2}{30/30}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {30}{30}}} +\HyPL@Entry{30<</P(25)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{3}{31/31}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {31}{31}}} +\HyPL@Entry{31<</P(26)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{4}{32/32}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {32}{32}}} +\HyPL@Entry{32<</P(27)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{5}{33/33}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {33}{33}}} +\HyPL@Entry{33<</P(28)>>} +\@writefile{nav}{\headcommand {\slideentry {8}{0}{6}{34/34}{}{0}}} +\@writefile{nav}{\headcommand {\beamer@framepages {34}{34}}} +\@writefile{nav}{\headcommand {\beamer@partpages {1}{34}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {29}{34}}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {29}{34}}} +\@writefile{nav}{\headcommand {\beamer@documentpages {34}}} +\@writefile{nav}{\headcommand {\gdef \inserttotalframenumber {28}}} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.log b/buch/papers/reedsolomon/RS presentation/RS_handout.log new file mode 100644 index 0000000..988a7c5 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.log @@ -0,0 +1,1198 @@ +This is XeTeX, Version 3.14159265-2.6-0.999991 (TeX Live 2019/W32TeX) (preloaded format=xelatex 2019.10.25) 24 APR 2021 15:25 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**RS_handout.tex +(./RS_handout.tex +LaTeX2e <2019-10-01> patch level 1 +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamer.cls +Document Class: beamer 2019/09/29 v3.57 A class for typesetting presentations +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasemodes.sty +(c:/texlive/2019/texmf-dist/tex/latex/etoolbox/etoolbox.sty +Package: etoolbox 2019/09/21 v2.5h e-TeX tools for LaTeX (JAW) +\etb@tempcnta=\count80 +) +\beamer@tempbox=\box27 +\beamer@tempcount=\count81 +\c@beamerpauses=\count82 + +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasedecode.sty +\beamer@slideinframe=\count83 +\beamer@minimum=\count84 +\beamer@decode@box=\box28 +) +\beamer@commentbox=\box29 +\beamer@modecount=\count85 +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch +) +\headdp=\dimen102 +\footheight=\dimen103 +\sidebarheight=\dimen104 +\beamer@tempdim=\dimen105 +\beamer@finalheight=\dimen106 +\beamer@animht=\dimen107 +\beamer@animdp=\dimen108 +\beamer@animwd=\dimen109 +\beamer@leftmargin=\dimen110 +\beamer@rightmargin=\dimen111 +\beamer@leftsidebar=\dimen112 +\beamer@rightsidebar=\dimen113 +\beamer@boxsize=\dimen114 +\beamer@vboxoffset=\dimen115 +\beamer@descdefault=\dimen116 +\beamer@descriptionwidth=\dimen117 +\beamer@lastskip=\skip41 +\beamer@areabox=\box30 +\beamer@animcurrent=\box31 +\beamer@animshowbox=\box32 +\beamer@sectionbox=\box33 +\beamer@logobox=\box34 +\beamer@linebox=\box35 +\beamer@sectioncount=\count86 +\beamer@subsubsectionmax=\count87 +\beamer@subsectionmax=\count88 +\beamer@sectionmax=\count89 +\beamer@totalheads=\count90 +\beamer@headcounter=\count91 +\beamer@partstartpage=\count92 +\beamer@sectionstartpage=\count93 +\beamer@subsectionstartpage=\count94 +\beamer@animationtempa=\count95 +\beamer@animationtempb=\count96 +\beamer@xpos=\count97 +\beamer@ypos=\count98 +\beamer@ypos@offset=\count99 +\beamer@showpartnumber=\count100 +\beamer@currentsubsection=\count101 +\beamer@coveringdepth=\count102 +\beamer@sectionadjust=\count103 +\beamer@tocsectionnumber=\count104 + +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseoptions.sty +(c:/texlive/2019/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 2014/10/28 v1.15 key=value parser (DPC) +\KV@toks@=\toks14 +)) +\beamer@paperwidth=\skip42 +\beamer@paperheight=\skip43 + +(c:/texlive/2019/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2018/04/16 v5.8 Page Geometry + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +) +(c:/texlive/2019/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +\Gm@cnth=\count105 +\Gm@cntv=\count106 +\c@Gm@tempcnt=\count107 +\Gm@bindingoffset=\dimen118 +\Gm@wd@mp=\dimen119 +\Gm@odd@mp=\dimen120 +\Gm@even@mp=\dimen121 +\Gm@layoutwidth=\dimen122 +\Gm@layoutheight=\dimen123 +\Gm@layouthoffset=\dimen124 +\Gm@layoutvoffset=\dimen125 +\Gm@dimlist=\toks15 +) +(c:/texlive/2019/texmf-dist/tex/latex/base/size11.clo +File: size11.clo 2019/08/27 v1.4j Standard LaTeX file (size option) +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2019/10/08 v1.3c Standard LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 2016/01/03 v1.10 sin cos tan (DPC) +) +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration +) +Package graphics Info: Driver file: xetex.def on input line 105. + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-def/xetex.def +File: xetex.def 2017/06/24 v5.0h Graphics/color driver for xetex +)) +\Gin@req@height=\dimen126 +\Gin@req@width=\dimen127 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks16 +\pgfutil@tempdima=\dimen128 +\pgfutil@tempdimb=\dimen129 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box36 + +(c:/texlive/2019/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/pgf.revision.tex) +Package: pgfrcs 2019/08/03 v3.1.4b (3.1.4b) +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks17 +\pgfkeys@temptoks=\toks18 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks19 +)) +\pgf@x=\dimen130 +\pgf@y=\dimen131 +\pgf@xa=\dimen132 +\pgf@ya=\dimen133 +\pgf@xb=\dimen134 +\pgf@yb=\dimen135 +\pgf@xc=\dimen136 +\pgf@yc=\dimen137 +\pgf@xd=\dimen138 +\pgf@yd=\dimen139 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count108 +\c@pgf@countb=\count109 +\c@pgf@countc=\count110 +\c@pgf@countd=\count111 +\t@pgf@toka=\toks20 +\t@pgf@tokb=\toks21 +\t@pgf@tokc=\toks22 +\pgf@sys@id@count=\count112 + (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2019/08/03 v3.1.4b (3.1.4b) +) +Driver file for pgf: pgfsys-xetex.def + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-xetex.def +File: pgfsys-xetex.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-dvipdfmx.def +File: pgfsys-dvipdfmx.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2019/08/03 v3.1.4b (3.1.4b) +) +\pgfsys@objnum=\count113 +))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfsyssoftpath@smallbuffer@items=\count114 +\pgfsyssoftpath@bigbuffer@items=\count115 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/xcolor/xcolor.sty +Package: xcolor 2016/05/11 v2.12 LaTeX color extensions (UK) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/color.cfg +File: color.cfg 2016/01/02 v1.6 sample color configuration +) +Package xcolor Info: Driver file: xetex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1348. +Package xcolor Info: Model `RGB' extended on input line 1364. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1366. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1367. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1368. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1369. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1370. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1371. +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen140 +\pgfmath@count=\count116 +\pgfmath@box=\box37 +\pgfmath@toks=\toks23 +\pgfmath@stack@operand=\toks24 +\pgfmath@stack@operation=\toks25 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric +.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.t +ex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.co +de.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithm +etics.code.tex))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count117 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@picminx=\dimen141 +\pgf@picmaxx=\dimen142 +\pgf@picminy=\dimen143 +\pgf@picmaxy=\dimen144 +\pgf@pathminx=\dimen145 +\pgf@pathmaxx=\dimen146 +\pgf@pathminy=\dimen147 +\pgf@pathmaxy=\dimen148 +\pgf@xx=\dimen149 +\pgf@xy=\dimen150 +\pgf@yx=\dimen151 +\pgf@yy=\dimen152 +\pgf@zx=\dimen153 +\pgf@zy=\dimen154 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.cod +e.tex +File: pgfcorepathconstruct.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@path@lastx=\dimen155 +\pgf@path@lasty=\dimen156 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.te +x +File: pgfcorepathusage.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@shorten@end@additional=\dimen157 +\pgf@shorten@start@additional=\dimen158 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfpic=\box38 +\pgf@hbox=\box39 +\pgf@layerbox@main=\box40 +\pgf@picture@serial@count=\count118 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code +.tex +File: pgfcoregraphicstate.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgflinewidth=\dimen159 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.c +ode.tex +File: pgfcoretransformations.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@pt@x=\dimen160 +\pgf@pt@y=\dimen161 +\pgf@pt@temp=\dimen162 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.co +de.tex +File: pgfcorepathprocessing.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfarrowsep=\dimen163 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@max=\dimen164 +\pgf@sys@shading@range@num=\count119 +\pgf@shadingcount=\count120 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfexternal@startupbox=\box41 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code +.tex +File: pgfcoretransparency.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +File: pgfcorerdf.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/xxcolor.sty +Package: xxcolor 2003/10/24 ver 0.1 +\XC@nummixins=\count121 +\XC@countmixins=\count122 +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/atbegshi.sty +Package: atbegshi 2016/06/09 v1.18 At begin shipout hook (HO) + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/infwarerr.sty +Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ltxcmds.sty +Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) +)) +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/hyperref.sty +Package: hyperref 2019/09/28 v7.00a Hypertext links for LaTeX + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty +Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) + + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty +Package: hobsub-generic 2016/05/16 v1.14 Bundle oberdiek, subset generic (HO) +Package: hobsub 2016/05/16 v1.14 Construct package bundles (HO) +Package hobsub Info: Skipping package `infwarerr' (already loaded). +Package hobsub Info: Skipping package `ltxcmds' (already loaded). +Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +Package hobsub Info: Skipping package `ifvtex' (already loaded). +Package: intcalc 2016/05/16 v1.2 Expandable calculations with integers (HO) +Package hobsub Info: Skipping package `ifpdf' (already loaded). +Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) +Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) +Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) +Package: pdftexcmds 2019/07/25 v0.30 Utility functions of pdfTeX for LuaTeX (HO +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: pdfTeX >= 1.30 not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode not found. +Package: pdfescape 2016/05/16 v1.14 Implements pdfTeX's escape features (HO) +Package: bigintcalc 2016/05/16 v1.4 Expandable calculations on big integers (HO +) +Package: bitset 2016/05/16 v1.2 Handle bit-vector datatype (HO) +Package: uniquecounter 2016/05/16 v1.3 Provide unlimited unique counter (HO) +) +Package hobsub Info: Skipping package `hobsub' (already loaded). +Package: letltxmacro 2016/05/16 v1.5 Let assignment for LaTeX macros (HO) +Package: hopatch 2016/05/16 v1.3 Wrapper for package hooks (HO) +Package: xcolor-patch 2016/05/16 xcolor patch +Package: atveryend 2016/05/16 v1.9 Hooks at the very end of document (HO) +Package hobsub Info: Skipping package `atbegshi' (already loaded). +Package: refcount 2016/05/16 v3.5 Data extraction from label references (HO) +Package: hycolor 2016/05/16 v1.8 Color options for hyperref/bookmark (HO) +) +(c:/texlive/2019/texmf-dist/tex/latex/oberdiek/auxhook.sty +Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) +) +(c:/texlive/2019/texmf-dist/tex/latex/oberdiek/kvoptions.sty +Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) +) +\@linkdim=\dimen165 +\Hy@linkcounter=\count123 +\Hy@pagecounter=\count124 + +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/pd1enc.def +File: pd1enc.def 2019/09/28 v7.00a Hyperref: PDFDocEncoding definition (HO) +) +\Hy@SavedSpaceFactor=\count125 + +(c:/texlive/2019/texmf-dist/tex/latex/latexconfig/hyperref.cfg +File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive +) +Package hyperref Info: Option `bookmarks' set `true' on input line 4414. +Package hyperref Info: Option `bookmarksopen' set `true' on input line 4414. +Package hyperref Info: Option `implicit' set `false' on input line 4414. +Package hyperref Info: Hyper figures OFF on input line 4540. +Package hyperref Info: Link nesting OFF on input line 4545. +Package hyperref Info: Hyper index ON on input line 4548. +Package hyperref Info: Plain pages OFF on input line 4555. +Package hyperref Info: Backreferencing OFF on input line 4560. +Package hyperref Info: Implicit mode OFF; no redefinition of LaTeX internals. +Package hyperref Info: Bookmarks ON on input line 4793. +\c@Hy@tempcnt=\count126 + +(c:/texlive/2019/texmf-dist/tex/latex/url/url.sty +\Urlmuskip=\muskip10 +Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. +) +LaTeX Info: Redefining \url on input line 5152. +\XeTeXLinkMargin=\dimen166 +\Fld@menulength=\count127 +\Field@Width=\dimen167 +\Fld@charsize=\dimen168 +Package hyperref Info: Hyper figures OFF on input line 6423. +Package hyperref Info: Link nesting OFF on input line 6428. +Package hyperref Info: Hyper index ON on input line 6431. +Package hyperref Info: backreferencing OFF on input line 6438. +Package hyperref Info: Link coloring OFF on input line 6443. +Package hyperref Info: Link coloring with OCG OFF on input line 6448. +Package hyperref Info: PDF/A mode OFF on input line 6453. +LaTeX Info: Redefining \ref on input line 6493. +LaTeX Info: Redefining \pageref on input line 6497. +\Hy@abspage=\count128 + + +Package hyperref Message: Stopped early. + +) +Package hyperref Info: Driver (autodetected): hxetex. + (c:/texlive/2019/texmf-dist/tex/latex/hyperref/hxetex.def +File: hxetex.def 2019/09/28 v7.00a Hyperref driver for XeTeX + +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/puenc.def +File: puenc.def 2019/09/28 v7.00a Hyperref: PDF Unicode definition (HO) +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/stringenc.sty +Package: stringenc 2016/05/16 v1.11 Convert strings between diff. encodings (HO +) +) +\pdfm@box=\box42 +\c@Hy@AnnotLevel=\count129 +\HyField@AnnotCount=\count130 +\Fld@listcount=\count131 +\c@bookmark@seq@number=\count132 + +(c:/texlive/2019/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty +Package: rerunfilecheck 2016/05/16 v1.8 Rerun checks for auxiliary files (HO) +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 +82. +)) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/se-ascii-print.def +File: se-ascii-print.def 2016/05/16 v1.11 stringenc: Printable ASCII characters + +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaserequires.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasecompatibility.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasefont.sty +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2013/01/14 v3.01 AMS font symbols + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\@emptytoks=\toks26 +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +)) +(c:/texlive/2019/texmf-dist/tex/latex/sansmathaccent/sansmathaccent.sty +Package: sansmathaccent 2013/03/28 + +(c:/texlive/2019/texmf-dist/tex/latex/filehook/filehook.sty +Package: filehook 2019/10/03 v0.6 Hooks for input files +))) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetranslator.sty +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator.sty +Package: translator 2019-05-31 v1.12a Easy translation of strings in LaTeX +)) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasemisc.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetwoscreens.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseoverlay.sty +\beamer@argscount=\count133 +\beamer@lastskipcover=\skip44 +\beamer@trivlistdepth=\count134 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetitle.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasesection.sty +\c@lecture=\count135 +\c@part=\count136 +\c@section=\count137 +\c@subsection=\count138 +\c@subsubsection=\count139 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseframe.sty +\beamer@framebox=\box43 +\beamer@frametitlebox=\box44 +\beamer@zoombox=\box45 +\beamer@zoomcount=\count140 +\beamer@zoomframecount=\count141 +\beamer@frametextheight=\dimen169 +\c@subsectionslide=\count142 +\beamer@frametopskip=\skip45 +\beamer@framebottomskip=\skip46 +\beamer@frametopskipautobreak=\skip47 +\beamer@framebottomskipautobreak=\skip48 +\beamer@envbody=\toks27 +\framewidth=\dimen170 +\c@framenumber=\count143 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseverbatim.sty +\beamer@verbatimfileout=\write4 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseframesize.sty +\beamer@splitbox=\box46 +\beamer@autobreakcount=\count144 +\beamer@autobreaklastheight=\dimen171 +\beamer@frametitletoks=\toks28 +\beamer@framesubtitletoks=\toks29 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseframecomponents.sty +\beamer@footins=\box47 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasecolor.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasenotes.sty +\beamer@frameboxcopy=\box48 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetoc.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetemplates.sty +\beamer@sbttoks=\toks30 + +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseauxtemplates.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaseboxes.sty +\bmb@box=\box49 +\bmb@colorbox=\box50 +\bmb@boxshadow=\box51 +\bmb@boxshadowball=\box52 +\bmb@boxshadowballlarge=\box53 +\bmb@temp=\dimen172 +\bmb@dima=\dimen173 +\bmb@dimb=\dimen174 +\bmb@prevheight=\dimen175 +) +\beamer@blockheadheight=\dimen176 +)) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbaselocalstructure.sty +(c:/texlive/2019/texmf-dist/tex/latex/tools/enumerate.sty +Package: enumerate 2015/07/23 v3.00 enumerate extensions (DPC) +\@enLab=\toks31 +) +\c@figure=\count145 +\c@table=\count146 +\abovecaptionskip=\skip49 +\belowcaptionskip=\skip50 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasenavigation.sty +\beamer@section@min@dim=\dimen177 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasetheorems.sty +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2019/04/01 v2.17c AMS math features +\@mathmargin=\skip51 + +For additional information on amsmath, use the `?' option. +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 AMS text + +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks32 +\ex@=\dimen178 +)) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen179 +) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 2016/03/08 v2.02 operator names +) +\inf@bad=\count147 +LaTeX Info: Redefining \frac on input line 227. +\uproot@=\count148 +\leftroot@=\count149 +LaTeX Info: Redefining \overline on input line 389. +\classnum@=\count150 +\DOTSCASE@=\count151 +LaTeX Info: Redefining \ldots on input line 486. +LaTeX Info: Redefining \dots on input line 489. +LaTeX Info: Redefining \cdots on input line 610. +\Mathstrutbox@=\box54 +\strutbox@=\box55 +\big@size=\dimen180 +LaTeX Font Info: Redeclaring font encoding OML on input line 733. +LaTeX Font Info: Redeclaring font encoding OMS on input line 734. +\macc@depth=\count152 +\c@MaxMatrixCols=\count153 +\dotsspace@=\muskip11 +\c@parentequation=\count154 +\dspbrk@lvl=\count155 +\tag@help=\toks33 +\row@=\count156 +\column@=\count157 +\maxfields@=\count158 +\andhelp@=\toks34 +\eqnshift@=\dimen181 +\alignsep@=\dimen182 +\tagshift@=\dimen183 +\tagwidth@=\dimen184 +\totwidth@=\dimen185 +\lineht@=\dimen186 +\@envbody=\toks35 +\multlinegap=\skip52 +\multlinetaggap=\skip53 +\mathdisplay@stack=\toks36 +LaTeX Info: Redefining \[ on input line 2855. +LaTeX Info: Redefining \] on input line 2856. +) +(c:/texlive/2019/texmf-dist/tex/latex/amscls/amsthm.sty +Package: amsthm 2017/10/31 v2.20.4 +\thm@style=\toks37 +\thm@bodyfont=\toks38 +\thm@headfont=\toks39 +\thm@notefont=\toks40 +\thm@headpunct=\toks41 +\thm@preskip=\skip54 +\thm@postskip=\skip55 +\thm@headsep=\skip56 +\dth@everypar=\toks42 +) +\c@theorem=\count159 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasethemes.sty)) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerthemedefault.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerfontthemedefault.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamercolorthemedefault.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerinnerthemedefault.sty +\beamer@dima=\dimen187 +\beamer@dimb=\dimen188 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerouterthemedefault.sty))) +(c:/texlive/2019/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2018/08/11 v1.3c Input encoding file +\inpenc@prehook=\toks43 +\inpenc@posthook=\toks44 + + +Package inputenc Warning: inputenc package ignored with utf8 based engines. + +) (c:/texlive/2019/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2018/08/11 v2.0j Standard LaTeX package + +(c:/texlive/2019/texmf-dist/tex/latex/base/t1enc.def +File: t1enc.def 2018/08/11 v2.0j Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 48. +) +LaTeX Font Info: Trying to load font information for T1+lmss on input line 1 +05. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/t1lmss.fd +File: t1lmss.fd 2009/10/30 v1.6 Font defs for Latin Modern +)) +(c:/texlive/2019/texmf-dist/tex/latex/lm/lmodern.sty +Package: lmodern 2009/10/30 v1.6 Latin Modern Fonts +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/cmr/m/n --> OT1/lmr/m/n on input line 22. +LaTeX Font Info: Overwriting symbol font `letters' in version `normal' +(Font) OML/cmm/m/it --> OML/lmm/m/it on input line 23. +LaTeX Font Info: Overwriting symbol font `symbols' in version `normal' +(Font) OMS/cmsy/m/n --> OMS/lmsy/m/n on input line 24. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `normal' +(Font) OMX/cmex/m/n --> OMX/lmex/m/n on input line 25. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/lmr/bx/n on input line 26. +LaTeX Font Info: Overwriting symbol font `letters' in version `bold' +(Font) OML/cmm/b/it --> OML/lmm/b/it on input line 27. +LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' +(Font) OMS/cmsy/b/n --> OMS/lmsy/b/n on input line 28. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' +(Font) OMX/cmex/m/n --> OMX/lmex/m/n on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/cmr/bx/n --> OT1/lmr/bx/n on input line 31. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' +(Font) OT1/cmss/m/n --> OT1/lmss/m/n on input line 32. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' +(Font) OT1/cmr/m/it --> OT1/lmr/m/it on input line 33. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' +(Font) OT1/cmtt/m/n --> OT1/lmtt/m/n on input line 34. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/lmr/bx/n on input line 35. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/cmss/bx/n --> OT1/lmss/bx/n on input line 36. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/cmr/bx/it --> OT1/lmr/bx/it on input line 37. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/cmtt/m/n --> OT1/lmtt/m/n on input line 38. +) +(c:/texlive/2019/texmf-dist/tex/generic/babel/babel.sty +Package: babel 2019/10/15 3.35 The Babel package + +(c:/texlive/2019/texmf-dist/tex/generic/babel/switch.def +File: switch.def 2019/10/15 3.35 Babel switching mechanism +) +(c:/texlive/2019/texmf-dist/tex/generic/babel-german/ngerman.ldf +Language: ngerman 2018/12/08 v2.11 German support for babel (post-1996 orthogra +phy) + +(c:/texlive/2019/texmf-dist/tex/generic/babel-german/ngermanb.ldf +Language: ngermanb 2018/12/08 v2.11 German support for babel (post-1996 orthogr +aphy) + +(c:/texlive/2019/texmf-dist/tex/generic/babel/babel.def +File: babel.def 2019/10/15 3.35 Babel common definitions +\babel@savecnt=\count160 +\U@D=\dimen189 + +(c:/texlive/2019/texmf-dist/tex/generic/babel/xebabel.def +(c:/texlive/2019/texmf-dist/tex/generic/babel/txtbabel.def)) +LaTeX Info: Redefining \textlatin on input line 2185. +\bbl@dirlevel=\count161 +) +Package babel Info: Making " an active character on input line 121. +))) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +Package: pgf 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +File: pgfmoduleshapes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfnodeparttextbox=\box56 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.st +y +Package: pgfcomp-version-0-65 2019/08/03 v3.1.4b (3.1.4b) +\pgf@nodesepstart=\dimen190 +\pgf@nodesepend=\dimen191 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.st +y +Package: pgfcomp-version-1-18 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/math/pgfmath.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex) +\pgffor@iter=\dimen192 +\pgffor@skip=\dimen193 +\pgffor@stack=\toks45 +\pgffor@toks=\toks46 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.co +de.tex +File: pgflibraryplothandlers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@plot@mark@count=\count162 +\pgfplotmarksize=\dimen194 +) +\tikz@lastx=\dimen195 +\tikz@lasty=\dimen196 +\tikz@lastxsaved=\dimen197 +\tikz@lastysaved=\dimen198 +\tikz@lastmovetox=\dimen199 +\tikz@lastmovetoy=\dimen256 +\tikzleveldistance=\dimen257 +\tikzsiblingdistance=\dimen258 +\tikz@figbox=\box57 +\tikz@figbox@bg=\box58 +\tikz@tempbox=\box59 +\tikz@tempbox@bg=\box60 +\tikztreelevel=\count163 +\tikznumberofchildren=\count164 +\tikznumberofcurrentchild=\count165 +\tikz@fig@count=\count166 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfmatrixcurrentrow=\count167 +\pgfmatrixcurrentcolumn=\count168 +\pgf@matrix@numberofcolumns=\count169 +) +\tikz@expandcount=\count170 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarytopaths.code.tex +File: tikzlibrarytopaths.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerthemeHannover.sty +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerouterthemesidebar.sty +\beamer@sidebarwidth=\dimen259 +\beamer@headheight=\dimen260 +) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamercolorthemeseahorse.sty) +(c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerinnerthemecircles.sty)) +(./RS_handout.aux) +\openout1 = `RS_handout.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 9. +LaTeX Font Info: ... okay on input line 9. + +*geometry* driver: auto-detecting +*geometry* detected driver: xetex +*geometry* verbose mode - [ preamble ] result: +* driver: xetex +* paper: custom +* layout: <same size as paper> +* layoutoffset:(h,v)=(0.0pt,0.0pt) +* modes: includehead includefoot +* h-part:(L,W,R)=(59.22636pt, 381.79135pt, 14.22636pt) +* v-part:(T,H,B)=(0.0pt, 256.0748pt, 0.0pt) +* \paperwidth=455.24408pt +* \paperheight=256.0748pt +* \textwidth=381.79135pt +* \textheight=227.62207pt +* \oddsidemargin=-13.04362pt +* \evensidemargin=-13.04362pt +* \topmargin=-72.26999pt +* \headheight=14.22636pt +* \headsep=0.0pt +* \topskip=11.0pt +* \footskip=14.22636pt +* \marginparwidth=4.0pt +* \marginparsep=10.0pt +* \columnsep=10.0pt +* \skip\footins=10.0pt plus 4.0pt minus 2.0pt +* \hoffset=0.0pt +* \voffset=0.0pt +* \mag=1000 +* \@twocolumnfalse +* \@twosidefalse +* \@mparswitchfalse +* \@reversemarginfalse +* (1in=72.27pt=25.4mm, 1cm=28.453pt) + +ABD: EveryShipout initializing macros +\AtBeginShipoutBox=\box61 +Package hyperref Info: Link coloring OFF on input line 9. + +(c:/texlive/2019/texmf-dist/tex/latex/hyperref/nameref.sty +Package: nameref 2019/09/16 v2.46 Cross-referencing by name of section + +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/gettitlestring.sty +Package: gettitlestring 2016/05/16 v1.5 Cleanup title references (HO) +) +\c@section@level=\count171 +) +LaTeX Info: Redefining \ref on input line 9. +LaTeX Info: Redefining \pageref on input line 9. +LaTeX Info: Redefining \nameref on input line 9. + +(./RS_handout.out) (./RS_handout.out) +\@outlinefile=\write5 +\openout5 = `RS_handout.out'. + +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/lmr/m/n --> OT1/cmss/m/n on input line 9. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/lmr/bx/n --> OT1/cmss/bx/n on input line 9. +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/cmss/m/n --> OT1/lmss/m/n on input line 9. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/cmss/bx/n --> OT1/lmss/bx/n on input line 9. +\symnumbers=\mathgroup6 +\sympureletters=\mathgroup7 +LaTeX Font Info: Overwriting math alphabet `\mathrm' in version `normal' +(Font) OT1/lmss/m/n --> T1/lmr/m/n on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/lmr/bx/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/lmr/bx/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathsf on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' +(Font) OT1/lmss/m/n --> T1/lmss/m/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/lmss/bx/n --> T1/lmss/m/n on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathit on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' +(Font) OT1/lmr/m/it --> T1/lmss/m/it on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/lmr/bx/it --> T1/lmss/m/it on input line 9. +LaTeX Font Info: Redeclaring math alphabet \mathtt on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' +(Font) OT1/lmtt/m/n --> T1/lmtt/m/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/lmtt/m/n --> T1/lmtt/m/n on input line 9. +LaTeX Font Info: Overwriting symbol font `numbers' in version `bold' +(Font) T1/lmss/m/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting symbol font `pureletters' in version `bold' +(Font) T1/lmss/m/it --> T1/lmss/bx/it on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathrm' in version `bold' +(Font) OT1/lmss/bx/n --> T1/lmr/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) T1/lmss/bx/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) T1/lmss/m/n --> T1/lmss/bx/n on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) T1/lmss/m/it --> T1/lmss/bx/it on input line 9. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) T1/lmtt/m/n --> T1/lmtt/bx/n on input line 9. + +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-basic-dictionary-En +glish.dict +Dictionary: translator-basic-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-bibliography-dictio +nary-English.dict +Dictionary: translator-bibliography-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-environment-diction +ary-English.dict +Dictionary: translator-environment-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-months-dictionary-E +nglish.dict +Dictionary: translator-months-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-numbers-dictionary- +English.dict +Dictionary: translator-numbers-dictionary, Language: English +) +(c:/texlive/2019/texmf-dist/tex/latex/translator/translator-theorem-dictionary- +English.dict +Dictionary: translator-theorem-dictionary, Language: English +) (./RS_handout.nav) + +Package hyperref Warning: Option `pdfauthor' has already been used, +(hyperref) setting the option has no effect on input line 10. + + +Package hyperref Warning: Option `pdfsubject' has already been used, +(hyperref) setting the option has no effect on input line 16. + +[1 + +] +LaTeX Font Info: Trying to load font information for OT1+lmss on input line +32. + (c:/texlive/2019/texmf-dist/tex/latex/lm/ot1lmss.fd +File: ot1lmss.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: Trying to load font information for OML+lmm on input line 3 +2. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/omllmm.fd +File: omllmm.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: Trying to load font information for OMS+lmsy on input line +32. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/omslmsy.fd +File: omslmsy.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: Trying to load font information for OMX+lmex on input line +32. + +(c:/texlive/2019/texmf-dist/tex/latex/lm/omxlmex.fd +File: omxlmex.fd 2009/10/30 v1.6 Font defs for Latin Modern +) +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <10.95> on input line 32. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <8> on input line 32. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <6> on input line 32. +LaTeX Font Info: Trying to load font information for U+msa on input line 32. + + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Trying to load font information for U+msb on input line 32. + + +(c:/texlive/2019/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <10.95> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 32. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <8> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 32. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <6> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 32. + [2 + +] +File: images/polynom2.pdf Graphic file (type pdf) +<use images/polynom2.pdf> + [3 + +] [4 + +] [5 + +] +File: images/fig1.pdf Graphic file (type pdf) +<use images/fig1.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[6 + +] +File: images/fig2.pdf Graphic file (type pdf) +<use images/fig2.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[7 + +] +File: images/fig3.pdf Graphic file (type pdf) +<use images/fig3.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[8 + +] +File: images/fig4.pdf Graphic file (type pdf) +<use images/fig4.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[9 + +] +File: images/fig5.pdf Graphic file (type pdf) +<use images/fig5.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[10 + +] +File: images/fig6.pdf Graphic file (type pdf) +<use images/fig6.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[11 + +] +File: images/fig7.pdf Graphic file (type pdf) +<use images/fig7.pdf> + +Overfull \vbox (14.63716pt too high) detected at line 111 + [] + +[12 + +] [13 + +] [14 + +] [15 + +] [16 + +] [17 + +] +LaTeX Font Info: Trying to load font information for T1+lmr on input line 28 +6. + (c:/texlive/2019/texmf-dist/tex/latex/lm/t1lmr.fd +File: t1lmr.fd 2009/10/30 v1.6 Font defs for Latin Modern +) [18 + +] [19 + +] +LaTeX Font Info: Trying to load font information for U+euf on input line 368 +. + (c:/texlive/2019/texmf-dist/tex/latex/amsfonts/ueuf.fd +File: ueuf.fd 2013/01/14 v3.01 Euler Fraktur +) [20 + +] +[21 + +] [22 + +] [23 + +] [24 + +] [25 + +] +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <14.4> on input line 618. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <10> on input line 618. +LaTeX Font Info: External font `lmex10' loaded for size +(Font) <7> on input line 618. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <14.4> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 618. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <10> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 618. +LaTeX Font Info: Font shape `T1/lmss/m/it' in size <7> not available +(Font) Font shape `T1/lmss/m/sl' tried instead on input line 618. + [26 + +] +Overfull \hbox (2.91844pt too wide) detected at line 653 +[] + [] + +[27 + +] [28 + +] [29 + +] [30 + +] [31 + +] [32 + +] [33 + +] [34 + +] +\tf@nav=\write6 +\openout6 = `RS_handout.nav'. + +\tf@toc=\write7 +\openout7 = `RS_handout.toc'. + +\tf@snm=\write8 +\openout8 = `RS_handout.snm'. + +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 907. +Package atveryend Info: Empty hook `AfterLastShipout' on input line 907. + (./RS_handout.aux) +Package atveryend Info: Empty hook `AtVeryEndDocument' on input line 907. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 907. +Package rerunfilecheck Info: File `RS_handout.out' has not changed. +(rerunfilecheck) Checksum: CBEDF1F633104E8EE4EB074E401487DA. + ) +Here is how much of TeX's memory you used: + 24519 strings out of 492483 + 453771 string characters out of 6132858 + 551669 words of memory out of 5000000 + 28480 multiletter control sequences out of 15000+600000 + 82614 words of font info for 73 fonts, out of 8000000 for 9000 + 1348 hyphenation exceptions out of 8191 + 58i,15n,61p,804b,549s stack positions out of 5000i,500n,10000p,200000b,80000s + +Output written on RS_handout.pdf (34 pages). diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.nav b/buch/papers/reedsolomon/RS presentation/RS_handout.nav new file mode 100644 index 0000000..b6e8a36 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.nav @@ -0,0 +1,85 @@ +\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}} +\headcommand {\beamer@framepages {1}{1}} +\headcommand {\beamer@sectionpages {1}{1}} +\headcommand {\beamer@subsectionpages {1}{1}} +\headcommand {\sectionentry {1}{Einführung}{2}{Einführung}{0}} +\headcommand {\slideentry {1}{0}{1}{2/2}{}{0}} +\headcommand {\beamer@framepages {2}{2}} +\headcommand {\beamer@sectionpages {2}{2}} +\headcommand {\beamer@subsectionpages {2}{2}} +\headcommand {\sectionentry {2}{Polynom Ansatz}{3}{Polynom Ansatz}{0}} +\headcommand {\slideentry {2}{0}{1}{3/3}{}{0}} +\headcommand {\beamer@framepages {3}{3}} +\headcommand {\slideentry {2}{0}{2}{4/4}{}{0}} +\headcommand {\beamer@framepages {4}{4}} +\headcommand {\beamer@sectionpages {3}{4}} +\headcommand {\beamer@subsectionpages {3}{4}} +\headcommand {\sectionentry {3}{Diskrete Fourier Transformation}{5}{Diskrete Fourier Transformation}{0}} +\headcommand {\slideentry {3}{0}{1}{5/5}{}{0}} +\headcommand {\beamer@framepages {5}{5}} +\headcommand {\slideentry {3}{0}{2}{6/12}{}{0}} +\headcommand {\beamer@framepages {6}{12}} +\headcommand {\slideentry {3}{0}{3}{13/13}{}{0}} +\headcommand {\beamer@framepages {13}{13}} +\headcommand {\slideentry {3}{0}{4}{14/14}{}{0}} +\headcommand {\beamer@framepages {14}{14}} +\headcommand {\slideentry {3}{0}{5}{15/15}{}{0}} +\headcommand {\beamer@framepages {15}{15}} +\headcommand {\beamer@sectionpages {5}{15}} +\headcommand {\beamer@subsectionpages {5}{15}} +\headcommand {\sectionentry {4}{Reed-Solomon in Endlichen Körpern}{16}{Reed-Solomon in Endlichen Körpern}{0}} +\headcommand {\slideentry {4}{0}{1}{16/16}{}{0}} +\headcommand {\beamer@framepages {16}{16}} +\headcommand {\slideentry {4}{0}{2}{17/17}{}{0}} +\headcommand {\beamer@framepages {17}{17}} +\headcommand {\beamer@sectionpages {16}{17}} +\headcommand {\beamer@subsectionpages {16}{17}} +\headcommand {\sectionentry {5}{Codierung eines Beispiels}{18}{Codierung eines Beispiels}{0}} +\headcommand {\slideentry {5}{0}{1}{18/18}{}{0}} +\headcommand {\beamer@framepages {18}{18}} +\headcommand {\slideentry {5}{0}{2}{19/19}{}{0}} +\headcommand {\beamer@framepages {19}{19}} +\headcommand {\beamer@sectionpages {18}{19}} +\headcommand {\beamer@subsectionpages {18}{19}} +\headcommand {\sectionentry {6}{Decodierung ohne Fehler}{20}{Decodierung ohne Fehler}{0}} +\headcommand {\slideentry {6}{0}{1}{20/20}{}{0}} +\headcommand {\beamer@framepages {20}{20}} +\headcommand {\slideentry {6}{0}{2}{21/21}{}{0}} +\headcommand {\beamer@framepages {21}{21}} +\headcommand {\slideentry {6}{0}{3}{22/22}{}{0}} +\headcommand {\beamer@framepages {22}{22}} +\headcommand {\beamer@sectionpages {20}{22}} +\headcommand {\beamer@subsectionpages {20}{22}} +\headcommand {\sectionentry {7}{Decodierung mit Fehler}{23}{Decodierung mit Fehler}{0}} +\headcommand {\slideentry {7}{0}{1}{23/23}{}{0}} +\headcommand {\beamer@framepages {23}{23}} +\headcommand {\slideentry {7}{0}{2}{24/24}{}{0}} +\headcommand {\beamer@framepages {24}{24}} +\headcommand {\slideentry {7}{0}{3}{25/25}{}{0}} +\headcommand {\beamer@framepages {25}{25}} +\headcommand {\slideentry {7}{0}{4}{26/26}{}{0}} +\headcommand {\beamer@framepages {26}{26}} +\headcommand {\slideentry {7}{0}{5}{27/27}{}{0}} +\headcommand {\beamer@framepages {27}{27}} +\headcommand {\slideentry {7}{0}{6}{28/28}{}{0}} +\headcommand {\beamer@framepages {28}{28}} +\headcommand {\beamer@sectionpages {23}{28}} +\headcommand {\beamer@subsectionpages {23}{28}} +\headcommand {\sectionentry {8}{Nachricht Rekonstruieren}{29}{Nachricht Rekonstruieren}{0}} +\headcommand {\slideentry {8}{0}{1}{29/29}{}{0}} +\headcommand {\beamer@framepages {29}{29}} +\headcommand {\slideentry {8}{0}{2}{30/30}{}{0}} +\headcommand {\beamer@framepages {30}{30}} +\headcommand {\slideentry {8}{0}{3}{31/31}{}{0}} +\headcommand {\beamer@framepages {31}{31}} +\headcommand {\slideentry {8}{0}{4}{32/32}{}{0}} +\headcommand {\beamer@framepages {32}{32}} +\headcommand {\slideentry {8}{0}{5}{33/33}{}{0}} +\headcommand {\beamer@framepages {33}{33}} +\headcommand {\slideentry {8}{0}{6}{34/34}{}{0}} +\headcommand {\beamer@framepages {34}{34}} +\headcommand {\beamer@partpages {1}{34}} +\headcommand {\beamer@subsectionpages {29}{34}} +\headcommand {\beamer@sectionpages {29}{34}} +\headcommand {\beamer@documentpages {34}} +\headcommand {\gdef \inserttotalframenumber {28}} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.out b/buch/papers/reedsolomon/RS presentation/RS_handout.out new file mode 100644 index 0000000..364319e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.out @@ -0,0 +1,8 @@ +\BOOKMARK [2][]{Outline0.1}{Einführung}{}% 1 +\BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2 +\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5 +\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6 +\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7 +\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8 diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.pdf b/buch/papers/reedsolomon/RS presentation/RS_handout.pdf Binary files differnew file mode 100644 index 0000000..382049d --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.pdf diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.snm b/buch/papers/reedsolomon/RS presentation/RS_handout.snm new file mode 100644 index 0000000..1796304 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.snm @@ -0,0 +1 @@ +\beamer@slide {ft_discrete}{13} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz Binary files differnew file mode 100644 index 0000000..c28a28a --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.tex b/buch/papers/reedsolomon/RS presentation/RS_handout.tex new file mode 100644 index 0000000..1cbb6ef --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.tex @@ -0,0 +1,907 @@ +\documentclass[11pt,aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[ngerman]{babel} +\usepackage{tikz} +\usetheme{Hannover} + +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Reed-Solomon-Code:} + \begin{itemize} + \item Für Übertragung von Daten + \item Ermöglicht Korrektur von Übertragungsfehler + \item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc. + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- +\section{Polynom Ansatz} + \begin{frame} + \begin{itemize} + \item $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} + \frametitle{Beispiel} + Übertragen von + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + \newline + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern. + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3& 3& 9 Werte eines Polynoms vom Grad 2 \\ + &&\\ + $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ + \hline + &&\\ + &&\\ + \multicolumn{3}{l} { + Ausserdem können bis zu $2t$ Fehler erkannt werden! + } + \end{tabular} + \end{center} + \end{frame} + +%------------------------------------------------------------------------------- + +\section{Diskrete Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \] + + \item Ersetzte + \[ + w = e^{-\frac{2\pi j}{N} k} + \] + + \item Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] + \end{itemize} + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = \frac{1}{N} + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] + \end{frame} +%------------------------------------------------------------------------------- + + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \newline + Indem in einem endlichen Körper gerechnet wird. + + \end{frame} + +%------------------------------------------------------------------------------- + + +\section{Reed-Solomon in Endlichen Körpern} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \item Warum endliche Körper? + + \qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler + + \qquad digitale Fehlerkorrektur + + %\onslide<4->{\qquad bessere Laufzeit} + + \vspace{10pt} + + \item Nachricht = Nutzdaten + Fehlerkorrekturteil + + \vspace{10pt} + + \item aus Fehlerkorrekturteil die Fehlerstellen finden + + \qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \item endlicher Körper $q = 11$ + + ist eine Primzahl + + beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen + + $n = q - 1 = 10$ Zahlen + + \vspace{10pt} + + \item Max.~Fehler $t = 2$ + + maximale Anzahl von Fehler, die wir noch korrigieren können + + \vspace{10pt} + + \item Nutzlast $k = n -2t = 6$ Zahlen + + Fehlerkorrkturstellen $2t = 4$ Zahlen + + Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$ + + als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation + + \vspace{10pt} + + \item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$ + + \vspace{10pt} + + \item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken + + $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$ + + \vspace{10pt} + + \item Wir wählen $a = 8$ + + $\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$ + + $8$ ist eine primitive Einheitswurzel + + \vspace{10pt} + + \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + + $\Rightarrow$ \qquad können wir auch als Matrix schreiben + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Übertragungsvektor $v$ + + \item $v = A \cdot m$ + + \end{itemize} + + \[ + v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$ + + \vspace{10pt} + + \item Wir suchen die Inverse der Matrix $A$ + + \vspace{10pt} + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.55\textwidth} + Inverse der Fouriertransformation + \vspace{10pt} + + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + + \vspace{10pt} + + \[ + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + + \end{column} + \begin{column}{0.45\textwidth} + Inverse von $a$ + + \vspace{10pt} + + \[ + 8^{1} \Rightarrow 8^{-1} + \] + + Inverse finden wir über den Eulkidischen Algorithmus + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + + \begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + + + \vspace{10pt} + + \begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Inverser Matrix} + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item $m = 1/10 \cdot A^{-1} \cdot v$ + + \item $m = 10 \cdot A^{-1} \cdot v$ + + \end{itemize} + + \[ + m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$ + + \end{itemize} + + Wie finden wir die Fehler? + + \begin{itemize} + \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + + \item $e(X) = r(X) - m(X)$ + + \end{itemize} + + \begin{center} + + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline + \end{tabular} + + \end{center} + + \begin{itemize} + \item Alle Stellen, die nicht Null sind, sind Fehler + \end{itemize} + + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + + \vspace{10pt} + + $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + + $= d(X) \cdot e(X)$ + + \vspace{10pt} + + \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Kennen wir $e(X)$?} + + \begin{itemize} + + \item $e(X)$ ist unbekannt auf der Empfängerseite + + \vspace{10pt} + + \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + + \vspace{10pt} + + \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + + \vspace{10pt} + + \item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + + \vspace{10pt} + + \item $f(X) = X^{10} - 1 = X^{10} + 10$ + + \vspace{10pt} + + \item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + $\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$ + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + + \vspace{10pt} + + $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ + Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ + Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \end{tabular} + + \vspace{10pt} + + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + + + $d(X) = (X-a^3)(X-a^8)$ + + \end{frame} +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item $d(X) = (X-\textcolor{red}{a^3})(X-\textcolor{red}{a^8})$ + + \end{itemize} + + \[ + \textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Fehlerstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{green}{8^6}& \textcolor{green}{8^7}& \textcolor{green}{8^8}& \textcolor{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Nullstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix in eine Quadratische Form bringen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix Invertieren + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + $\Downarrow$ + \end{center} + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + +\end{document} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.toc b/buch/papers/reedsolomon/RS presentation/RS_handout.toc new file mode 100644 index 0000000..ce1bdc2 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.toc @@ -0,0 +1,9 @@ +\babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{5}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{16}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{18}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{20}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{23}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{29}{0}{8} diff --git a/buch/papers/reedsolomon/RS presentation/images/fig1.pdf b/buch/papers/reedsolomon/RS presentation/images/fig1.pdf Binary files differnew file mode 100644 index 0000000..abde60c --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig1.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig1.svg b/buch/papers/reedsolomon/RS presentation/images/fig1.svg new file mode 100644 index 0000000..8682b56 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig1.svg @@ -0,0 +1,180 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ><font horiz-adv-x="75.0" id="font1" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:normal;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="0" horiz-adv-x="55.615234" d="M4.1562 35.2969 Q4.1562 48 6.7656 55.7422 Q9.375 63.4844 14.5234 67.6797 Q19.6719 71.875 27.4844 71.875 Q33.25 71.875 37.5938 69.5547 Q41.9375 67.2344 44.7734 62.8672 Q47.6094 58.5 49.2188 52.2266 Q50.8281 45.9531 50.8281 35.2969 Q50.8281 22.7031 48.2422 14.9688 Q45.6562 7.2344 40.5078 3.0078 Q35.3594 -1.2188 27.4844 -1.2188 Q17.1406 -1.2188 11.2344 6.2031 Q4.1562 15.1406 4.1562 35.2969 ZM13.1875 35.2969 Q13.1875 17.6719 17.3125 11.8359 Q21.4375 6 27.4844 6 Q33.5469 6 37.6719 11.8594 Q41.7969 17.7188 41.7969 35.2969 Q41.7969 52.9844 37.6719 58.7891 Q33.5469 64.5938 27.3906 64.5938 Q21.3438 64.5938 17.7188 59.4688 Q13.1875 52.9375 13.1875 35.2969 Z" + /><glyph unicode="1" horiz-adv-x="55.615234" d="M37.25 0 L28.4688 0 L28.4688 56 Q25.2969 52.9844 20.1406 49.9531 Q14.9844 46.9219 10.8906 45.4062 L10.8906 53.9062 Q18.2656 57.375 23.7812 62.3047 Q29.2969 67.2344 31.5938 71.875 L37.25 71.875 L37.25 0 Z" + /><glyph unicode="2" horiz-adv-x="55.615234" d="M50.3438 8.4531 L50.3438 0 L3.0312 0 Q2.9375 3.1719 4.0469 6.1094 Q5.8594 10.9375 9.8359 15.625 Q13.8125 20.3125 21.3438 26.4688 Q33.0156 36.0312 37.1172 41.625 Q41.2188 47.2188 41.2188 52.2031 Q41.2188 57.4219 37.4766 61.0078 Q33.7344 64.5938 27.7344 64.5938 Q21.3906 64.5938 17.5781 60.7891 Q13.7656 56.9844 13.7188 50.25 L4.6875 51.1719 Q5.6094 61.2812 11.6641 66.5781 Q17.7188 71.875 27.9375 71.875 Q38.2344 71.875 44.2422 66.1641 Q50.25 60.4531 50.25 52 Q50.25 47.7031 48.4922 43.5547 Q46.7344 39.4062 42.6562 34.8125 Q38.5781 30.2188 29.1094 22.2188 Q21.1875 15.5781 18.9453 13.2109 Q16.7031 10.8438 15.2344 8.4531 L50.3438 8.4531 Z" + /><glyph unicode="3" horiz-adv-x="55.615234" d="M4.2031 18.8906 L12.9844 20.0625 Q14.5 12.5938 18.1406 9.2969 Q21.7812 6 27 6 Q33.2031 6 37.4766 10.2969 Q41.75 14.5938 41.75 20.9531 Q41.75 27 37.7969 30.9297 Q33.8438 34.8594 27.7344 34.8594 Q25.25 34.8594 21.5312 33.8906 L22.5156 41.6094 Q23.3906 41.5 23.9219 41.5 Q29.5469 41.5 34.0391 44.4297 Q38.5312 47.3594 38.5312 53.4688 Q38.5312 58.2969 35.2578 61.4766 Q31.9844 64.6562 26.8125 64.6562 Q21.6875 64.6562 18.2656 61.4297 Q14.8438 58.2031 13.875 51.7656 L5.0781 53.3281 Q6.6875 62.1562 12.3984 67.0156 Q18.1094 71.875 26.6094 71.875 Q32.4688 71.875 37.3984 69.3594 Q42.3281 66.8438 44.9453 62.5 Q47.5625 58.1562 47.5625 53.2656 Q47.5625 48.6406 45.0703 44.8281 Q42.5781 41.0156 37.7031 38.7656 Q44.0469 37.3125 47.5625 32.6953 Q51.0781 28.0781 51.0781 21.1406 Q51.0781 11.7656 44.2422 5.25 Q37.4062 -1.2656 26.9531 -1.2656 Q17.5312 -1.2656 11.3047 4.3516 Q5.0781 9.9688 4.2031 18.8906 Z" + /><glyph unicode="4" horiz-adv-x="55.615234" d="M32.3281 0 L32.3281 17.1406 L1.2656 17.1406 L1.2656 25.2031 L33.9375 71.5781 L41.1094 71.5781 L41.1094 25.2031 L50.7812 25.2031 L50.7812 17.1406 L41.1094 17.1406 L41.1094 0 L32.3281 0 ZM32.3281 25.2031 L32.3281 57.4688 L9.9062 25.2031 L32.3281 25.2031 Z" + /><glyph unicode="5" horiz-adv-x="55.615234" d="M4.1562 18.75 L13.375 19.5312 Q14.4062 12.7969 18.1406 9.3984 Q21.875 6 27.1562 6 Q33.5 6 37.8906 10.7891 Q42.2812 15.5781 42.2812 23.4844 Q42.2812 31 38.0625 35.3516 Q33.8438 39.7031 27 39.7031 Q22.75 39.7031 19.3359 37.7734 Q15.9219 35.8438 13.9688 32.7656 L5.7188 33.8438 L12.6406 70.6094 L48.25 70.6094 L48.25 62.2031 L19.6719 62.2031 L15.8281 42.9688 Q22.2656 47.4688 29.3438 47.4688 Q38.7188 47.4688 45.1641 40.9688 Q51.6094 34.4688 51.6094 24.2656 Q51.6094 14.5469 45.9531 7.4688 Q39.0625 -1.2188 27.1562 -1.2188 Q17.3906 -1.2188 11.2109 4.25 Q5.0312 9.7188 4.1562 18.75 Z" + /><glyph unicode="6" horiz-adv-x="55.615234" d="M49.75 54.0469 L41.0156 53.375 Q39.8438 58.5469 37.7031 60.8906 Q34.125 64.6562 28.9062 64.6562 Q24.7031 64.6562 21.5312 62.3125 Q17.3906 59.2812 14.9922 53.4688 Q12.5938 47.6562 12.5 36.9219 Q15.6719 41.75 20.2656 44.0938 Q24.8594 46.4375 29.8906 46.4375 Q38.6719 46.4375 44.8516 39.9688 Q51.0312 33.5 51.0312 23.25 Q51.0312 16.5 48.125 10.7188 Q45.2188 4.9375 40.1406 1.8594 Q35.0625 -1.2188 28.6094 -1.2188 Q17.625 -1.2188 10.6953 6.8594 Q3.7656 14.9375 3.7656 33.5 Q3.7656 54.25 11.4219 63.6719 Q18.1094 71.875 29.4375 71.875 Q37.8906 71.875 43.2891 67.1406 Q48.6875 62.4062 49.75 54.0469 ZM13.875 23.1875 Q13.875 18.6562 15.7969 14.5078 Q17.7188 10.3594 21.1875 8.1797 Q24.6562 6 28.4688 6 Q34.0312 6 38.0391 10.4922 Q42.0469 14.9844 42.0469 22.7031 Q42.0469 30.125 38.0859 34.3984 Q34.125 38.6719 28.125 38.6719 Q22.1719 38.6719 18.0234 34.3984 Q13.875 30.125 13.875 23.1875 Z" + /><glyph unicode="7" horiz-adv-x="55.615234" d="M4.7344 62.2031 L4.7344 70.6562 L51.0781 70.6562 L51.0781 63.8125 Q44.2344 56.5469 37.5234 44.4844 Q30.8125 32.4219 27.1562 19.6719 Q24.5156 10.6875 23.7812 0 L14.75 0 Q14.8906 8.4531 18.0625 20.4141 Q21.2344 32.375 27.1719 43.4844 Q33.1094 54.5938 39.7969 62.2031 L4.7344 62.2031 Z" + /><glyph unicode="8" horiz-adv-x="55.615234" d="M17.6719 38.8125 Q12.2031 40.8281 9.5703 44.5391 Q6.9375 48.25 6.9375 53.4219 Q6.9375 61.2344 12.5547 66.5547 Q18.1719 71.875 27.4844 71.875 Q36.8594 71.875 42.5781 66.4297 Q48.2969 60.9844 48.2969 53.1719 Q48.2969 48.1875 45.6797 44.5078 Q43.0625 40.8281 37.75 38.8125 Q44.3438 36.6719 47.7812 31.8828 Q51.2188 27.0938 51.2188 20.4531 Q51.2188 11.2812 44.7266 5.0312 Q38.2344 -1.2188 27.6406 -1.2188 Q17.0469 -1.2188 10.5469 5.0547 Q4.0469 11.3281 4.0469 20.7031 Q4.0469 27.6875 7.5938 32.3984 Q11.1406 37.1094 17.6719 38.8125 ZM15.9219 53.7188 Q15.9219 48.6406 19.1953 45.4141 Q22.4688 42.1875 27.6875 42.1875 Q32.7656 42.1875 36.0156 45.3828 Q39.2656 48.5781 39.2656 53.2188 Q39.2656 58.0625 35.9141 61.3594 Q32.5625 64.6562 27.5938 64.6562 Q22.5625 64.6562 19.2422 61.4297 Q15.9219 58.2031 15.9219 53.7188 ZM13.0938 20.6562 Q13.0938 16.8906 14.875 13.375 Q16.6562 9.8594 20.1719 7.9297 Q23.6875 6 27.7344 6 Q34.0312 6 38.1328 10.0547 Q42.2344 14.1094 42.2344 20.3594 Q42.2344 26.7031 38.0156 30.8594 Q33.7969 35.0156 27.4375 35.0156 Q21.2344 35.0156 17.1641 30.9141 Q13.0938 26.8125 13.0938 20.6562 Z" + /><glyph unicode="9" horiz-adv-x="55.615234" d="M5.4688 16.5469 L13.9219 17.3281 Q14.9844 11.375 18.0156 8.6875 Q21.0469 6 25.7812 6 Q29.8281 6 32.8828 7.8594 Q35.9375 9.7188 37.8906 12.8203 Q39.8438 15.9219 41.1641 21.1953 Q42.4844 26.4688 42.4844 31.9375 Q42.4844 32.5156 42.4375 33.6875 Q39.7969 29.5 35.2344 26.8828 Q30.6719 24.2656 25.3438 24.2656 Q16.4531 24.2656 10.3047 30.7109 Q4.1562 37.1562 4.1562 47.7031 Q4.1562 58.5938 10.5781 65.2344 Q17 71.875 26.6562 71.875 Q33.6406 71.875 39.4297 68.1172 Q45.2188 64.3594 48.2188 57.3984 Q51.2188 50.4375 51.2188 37.25 Q51.2188 23.5312 48.2422 15.4062 Q45.2656 7.2812 39.3828 3.0312 Q33.5 -1.2188 25.5938 -1.2188 Q17.1875 -1.2188 11.8672 3.4453 Q6.5469 8.1094 5.4688 16.5469 ZM41.4531 48.1406 Q41.4531 55.7188 37.4297 60.1562 Q33.4062 64.5938 27.7344 64.5938 Q21.875 64.5938 17.5312 59.8125 Q13.1875 55.0312 13.1875 47.4062 Q13.1875 40.5781 17.3125 36.3047 Q21.4375 32.0312 27.4844 32.0312 Q33.5938 32.0312 37.5234 36.3047 Q41.4531 40.5781 41.4531 48.1406 Z" + /></font + ><font horiz-adv-x="75.0" id="font2" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:bold;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="l" horiz-adv-x="27.783203" d="M7.1719 0 L7.1719 71.5781 L20.9062 71.5781 L20.9062 0 L7.1719 0 Z" + /><glyph unicode="a" horiz-adv-x="55.615234" d="M17.4375 36.0312 L4.9844 38.2812 Q7.0781 45.7969 12.2031 49.4141 Q17.3281 53.0312 27.4375 53.0312 Q36.625 53.0312 41.1172 50.8594 Q45.6094 48.6875 47.4375 45.3438 Q49.2656 42 49.2656 33.0625 L49.125 17.0469 Q49.125 10.2031 49.7812 6.9609 Q50.4375 3.7188 52.25 0 L38.6719 0 Q38.1406 1.375 37.3594 4.0469 Q37.0156 5.2812 36.8594 5.6719 Q33.3438 2.25 29.3438 0.5391 Q25.3438 -1.1719 20.7969 -1.1719 Q12.7969 -1.1719 8.1797 3.1719 Q3.5625 7.5156 3.5625 14.1562 Q3.5625 18.5625 5.6641 22 Q7.7656 25.4375 11.5469 27.2734 Q15.3281 29.1094 22.4688 30.4688 Q32.0781 32.2812 35.7969 33.8438 L35.7969 35.2031 Q35.7969 39.1562 33.8438 40.8438 Q31.8906 42.5312 26.4688 42.5312 Q22.7969 42.5312 20.75 41.0938 Q18.7031 39.6562 17.4375 36.0312 ZM35.7969 24.9062 Q33.1562 24.0312 27.4453 22.8047 Q21.7344 21.5781 19.9688 20.4062 Q17.2812 18.5 17.2812 15.5781 Q17.2812 12.7031 19.4297 10.6016 Q21.5781 8.5 24.9062 8.5 Q28.6094 8.5 31.9844 10.9375 Q34.4688 12.7969 35.25 15.4844 Q35.7969 17.2344 35.7969 22.1719 L35.7969 24.9062 Z" + /><glyph unicode="n" horiz-adv-x="61.083984" d="M54.3438 0 L40.625 0 L40.625 26.4688 Q40.625 34.8594 39.75 37.3281 Q38.875 39.7969 36.8906 41.1641 Q34.9062 42.5312 32.125 42.5312 Q28.5625 42.5312 25.7344 40.5781 Q22.9062 38.625 21.8516 35.3984 Q20.7969 32.1719 20.7969 23.4844 L20.7969 0 L7.0781 0 L7.0781 51.8594 L19.8281 51.8594 L19.8281 44.2344 Q26.6094 53.0312 36.9219 53.0312 Q41.4531 53.0312 45.2109 51.3906 Q48.9688 49.75 50.8984 47.2109 Q52.8281 44.6719 53.5859 41.4531 Q54.3438 38.2344 54.3438 32.2344 L54.3438 0 Z" + /><glyph unicode="g" horiz-adv-x="61.083984" d="M5.9062 -3.4219 L21.5781 -5.3281 Q21.9688 -8.0625 23.3906 -9.0781 Q25.3438 -10.5469 29.5469 -10.5469 Q34.9062 -10.5469 37.5938 -8.9375 Q39.4062 -7.8594 40.3281 -5.4688 Q40.9688 -3.7656 40.9688 0.8281 L40.9688 8.4062 Q34.8125 0 25.4375 0 Q14.9844 0 8.8906 8.8438 Q4.1094 15.8281 4.1094 26.2188 Q4.1094 39.2656 10.3828 46.1484 Q16.6562 53.0312 25.9844 53.0312 Q35.5938 53.0312 41.8438 44.5781 L41.8438 51.8594 L54.6875 51.8594 L54.6875 5.3281 Q54.6875 -3.8594 53.1719 -8.3984 Q51.6562 -12.9375 48.9219 -15.5234 Q46.1875 -18.1094 41.625 -19.5781 Q37.0625 -21.0469 30.0781 -21.0469 Q16.8906 -21.0469 11.375 -16.5312 Q5.8594 -12.0156 5.8594 -5.0781 Q5.8594 -4.3906 5.9062 -3.4219 ZM18.1719 27 Q18.1719 18.75 21.3672 14.9141 Q24.5625 11.0781 29.25 11.0781 Q34.2812 11.0781 37.75 15.0156 Q41.2188 18.9531 41.2188 26.6562 Q41.2188 34.7188 37.8984 38.625 Q34.5781 42.5312 29.5 42.5312 Q24.5625 42.5312 21.3672 38.6953 Q18.1719 34.8594 18.1719 27 Z" + /><glyph unicode="i" horiz-adv-x="27.783203" d="M7.1719 58.8906 L7.1719 71.5781 L20.9062 71.5781 L20.9062 58.8906 L7.1719 58.8906 ZM7.1719 0 L7.1719 51.8594 L20.9062 51.8594 L20.9062 0 L7.1719 0 Z" + /><glyph unicode="S" horiz-adv-x="66.69922" d="M3.6094 23.2969 L17.6719 24.6562 Q18.9531 17.5781 22.8281 14.2578 Q26.7031 10.9375 33.2969 10.9375 Q40.2812 10.9375 43.8203 13.8906 Q47.3594 16.8438 47.3594 20.7969 Q47.3594 23.3438 45.875 25.125 Q44.3906 26.9062 40.6719 28.2188 Q38.1406 29.1094 29.1094 31.3438 Q17.4844 34.2344 12.7969 38.4219 Q6.2031 44.3438 6.2031 52.8281 Q6.2031 58.2969 9.3047 63.0625 Q12.4062 67.8281 18.2422 70.3125 Q24.0781 72.7969 32.3281 72.7969 Q45.7969 72.7969 52.6094 66.8906 Q59.4219 60.9844 59.7656 51.125 L45.3125 50.4844 Q44.3906 56 41.3359 58.4219 Q38.2812 60.8438 32.1719 60.8438 Q25.875 60.8438 22.3125 58.25 Q20.0156 56.5938 20.0156 53.8125 Q20.0156 51.2656 22.1719 49.4688 Q24.9062 47.1719 35.4531 44.6797 Q46 42.1875 51.0547 39.5234 Q56.1094 36.8594 58.9609 32.25 Q61.8125 27.6406 61.8125 20.8438 Q61.8125 14.7031 58.3984 9.3281 Q54.9844 3.9531 48.7344 1.3438 Q42.4844 -1.2656 33.1562 -1.2656 Q19.5781 -1.2656 12.3047 5.0078 Q5.0312 11.2812 3.6094 23.2969 Z" + /></font + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >10</text + ></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >20</text + ></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >30</text + ></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >40</text + ></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >50</text + ></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >60</text + ></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >70</text + ></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >80</text + ></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >90</text + ></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="339.7" style="fill:none;" x1="73" x2="77.34" y1="339.7" + /><line y2="305.4" style="fill:none;" x1="73" x2="77.34" y1="305.4" + /><line y2="271.1" style="fill:none;" x1="73" x2="77.34" y1="271.1" + /><line y2="236.8" style="fill:none;" x1="73" x2="77.34" y1="236.8" + /><line y2="202.5" style="fill:none;" x1="73" x2="77.34" y1="202.5" + /><line y2="168.2" style="fill:none;" x1="73" x2="77.34" y1="168.2" + /><line y2="133.9" style="fill:none;" x1="73" x2="77.34" y1="133.9" + /><line y2="99.6" style="fill:none;" x1="73" x2="77.34" y1="99.6" + /><line y2="65.3" style="fill:none;" x1="73" x2="77.34" y1="65.3" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="339.7" style="fill:none;" x1="507" x2="502.66" y1="339.7" + /><line y2="305.4" style="fill:none;" x1="507" x2="502.66" y1="305.4" + /><line y2="271.1" style="fill:none;" x1="507" x2="502.66" y1="271.1" + /><line y2="236.8" style="fill:none;" x1="507" x2="502.66" y1="236.8" + /><line y2="202.5" style="fill:none;" x1="507" x2="502.66" y1="202.5" + /><line y2="168.2" style="fill:none;" x1="507" x2="502.66" y1="168.2" + /><line y2="133.9" style="fill:none;" x1="507" x2="502.66" y1="133.9" + /><line y2="99.6" style="fill:none;" x1="507" x2="502.66" y1="99.6" + /><line y2="65.3" style="fill:none;" x1="507" x2="502.66" y1="65.3" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >0</text + ></g + ><g transform="translate(67.6667,339.7)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >1</text + ></g + ><g transform="translate(67.6667,305.4)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >2</text + ></g + ><g transform="translate(67.6667,271.1)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >3</text + ></g + ><g transform="translate(67.6667,236.8)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >4</text + ></g + ><g transform="translate(67.6667,202.5)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >5</text + ></g + ><g transform="translate(67.6667,168.2)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >6</text + ></g + ><g transform="translate(67.6667,133.9)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >7</text + ></g + ><g transform="translate(67.6667,99.6)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >8</text + ></g + ><g transform="translate(67.6667,65.3)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >9</text + ></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-15" xml:space="preserve" y="5.5" style="stroke:none;" + >10</text + ></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><text x="-22" xml:space="preserve" y="-4" style="stroke:none;" + >Signal</text + ></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 202.5 L77.5684 236.8 L82.1368 236.8 L86.7053 271.1 L91.2737 202.5 L95.8421 202.5 L100.4105 99.6 L104.979 99.6 L109.5474 168.2 L114.1158 236.8 L118.6842 99.6 L123.2526 202.5 L127.8211 236.8 L132.3895 65.3 L136.9579 65.3 L141.5263 168.2 L146.0947 168.2 L150.6632 168.2 L155.2316 305.4 L159.8 271.1 L164.3684 202.5 L168.9368 305.4 L173.5053 99.6 L178.0737 305.4 L182.6421 305.4 L187.2105 305.4 L191.7789 305.4 L196.3474 236.8 L200.9158 271.1 L205.4842 65.3 L210.0526 236.8 L214.621 305.4 L219.1895 65.3 L223.7579 31 L228.3263 236.8 L232.8947 339.7 L237.4632 271.1 L242.0316 236.8 L246.6 168.2 L251.1684 271.1 L255.7368 168.2 L260.3053 133.9 L264.8737 305.4 L269.4421 339.7 L274.0105 271.1 L278.5789 271.1 L283.1474 236.8 L287.7158 202.5 L292.2842 339.7 L296.8526 271.1 L301.4211 99.6 L305.9895 374 L310.5579 65.3 L315.1263 133.9 L319.6947 202.5 L324.2632 168.2 L328.8316 305.4 L333.4 202.5 L337.9684 31 L342.5368 202.5 L347.1053 202.5 L351.6737 305.4 L356.2421 202.5 L360.8105 168.2 L365.3789 374 L369.9474 374 L374.5158 374 L379.0842 374 L383.6526 374 L388.221 374 L392.7895 374 L397.3579 374 L401.9263 374 L406.4948 374 L411.0632 374 L415.6316 374 L420.2 374 L424.7684 374 L429.3369 374 L433.9053 374 L438.4737 374 L443.0421 374 L447.6105 374 L452.179 374 L456.7474 374 L461.3158 374 L465.8842 374 L470.4526 374 L475.0211 374 L479.5895 374 L484.1579 374 L488.7263 374 L493.2947 374 L497.8632 374 L502.4316 374 L507 374" style="fill:none; fill-rule:evenodd;" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/fig2.pdf b/buch/papers/reedsolomon/RS presentation/images/fig2.pdf Binary files differnew file mode 100644 index 0000000..8281ce7 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig2.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig2.svg b/buch/papers/reedsolomon/RS presentation/images/fig2.svg new file mode 100644 index 0000000..e66bd95 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig2.svg @@ -0,0 +1,163 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ><font horiz-adv-x="75.0" id="font1" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:normal;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="0" horiz-adv-x="55.615234" d="M4.1562 35.2969 Q4.1562 48 6.7656 55.7422 Q9.375 63.4844 14.5234 67.6797 Q19.6719 71.875 27.4844 71.875 Q33.25 71.875 37.5938 69.5547 Q41.9375 67.2344 44.7734 62.8672 Q47.6094 58.5 49.2188 52.2266 Q50.8281 45.9531 50.8281 35.2969 Q50.8281 22.7031 48.2422 14.9688 Q45.6562 7.2344 40.5078 3.0078 Q35.3594 -1.2188 27.4844 -1.2188 Q17.1406 -1.2188 11.2344 6.2031 Q4.1562 15.1406 4.1562 35.2969 ZM13.1875 35.2969 Q13.1875 17.6719 17.3125 11.8359 Q21.4375 6 27.4844 6 Q33.5469 6 37.6719 11.8594 Q41.7969 17.7188 41.7969 35.2969 Q41.7969 52.9844 37.6719 58.7891 Q33.5469 64.5938 27.3906 64.5938 Q21.3438 64.5938 17.7188 59.4688 Q13.1875 52.9375 13.1875 35.2969 Z" + /><glyph unicode="1" horiz-adv-x="55.615234" d="M37.25 0 L28.4688 0 L28.4688 56 Q25.2969 52.9844 20.1406 49.9531 Q14.9844 46.9219 10.8906 45.4062 L10.8906 53.9062 Q18.2656 57.375 23.7812 62.3047 Q29.2969 67.2344 31.5938 71.875 L37.25 71.875 L37.25 0 Z" + /><glyph unicode="2" horiz-adv-x="55.615234" d="M50.3438 8.4531 L50.3438 0 L3.0312 0 Q2.9375 3.1719 4.0469 6.1094 Q5.8594 10.9375 9.8359 15.625 Q13.8125 20.3125 21.3438 26.4688 Q33.0156 36.0312 37.1172 41.625 Q41.2188 47.2188 41.2188 52.2031 Q41.2188 57.4219 37.4766 61.0078 Q33.7344 64.5938 27.7344 64.5938 Q21.3906 64.5938 17.5781 60.7891 Q13.7656 56.9844 13.7188 50.25 L4.6875 51.1719 Q5.6094 61.2812 11.6641 66.5781 Q17.7188 71.875 27.9375 71.875 Q38.2344 71.875 44.2422 66.1641 Q50.25 60.4531 50.25 52 Q50.25 47.7031 48.4922 43.5547 Q46.7344 39.4062 42.6562 34.8125 Q38.5781 30.2188 29.1094 22.2188 Q21.1875 15.5781 18.9453 13.2109 Q16.7031 10.8438 15.2344 8.4531 L50.3438 8.4531 Z" + /><glyph unicode="3" horiz-adv-x="55.615234" d="M4.2031 18.8906 L12.9844 20.0625 Q14.5 12.5938 18.1406 9.2969 Q21.7812 6 27 6 Q33.2031 6 37.4766 10.2969 Q41.75 14.5938 41.75 20.9531 Q41.75 27 37.7969 30.9297 Q33.8438 34.8594 27.7344 34.8594 Q25.25 34.8594 21.5312 33.8906 L22.5156 41.6094 Q23.3906 41.5 23.9219 41.5 Q29.5469 41.5 34.0391 44.4297 Q38.5312 47.3594 38.5312 53.4688 Q38.5312 58.2969 35.2578 61.4766 Q31.9844 64.6562 26.8125 64.6562 Q21.6875 64.6562 18.2656 61.4297 Q14.8438 58.2031 13.875 51.7656 L5.0781 53.3281 Q6.6875 62.1562 12.3984 67.0156 Q18.1094 71.875 26.6094 71.875 Q32.4688 71.875 37.3984 69.3594 Q42.3281 66.8438 44.9453 62.5 Q47.5625 58.1562 47.5625 53.2656 Q47.5625 48.6406 45.0703 44.8281 Q42.5781 41.0156 37.7031 38.7656 Q44.0469 37.3125 47.5625 32.6953 Q51.0781 28.0781 51.0781 21.1406 Q51.0781 11.7656 44.2422 5.25 Q37.4062 -1.2656 26.9531 -1.2656 Q17.5312 -1.2656 11.3047 4.3516 Q5.0781 9.9688 4.2031 18.8906 Z" + /><glyph unicode="4" horiz-adv-x="55.615234" d="M32.3281 0 L32.3281 17.1406 L1.2656 17.1406 L1.2656 25.2031 L33.9375 71.5781 L41.1094 71.5781 L41.1094 25.2031 L50.7812 25.2031 L50.7812 17.1406 L41.1094 17.1406 L41.1094 0 L32.3281 0 ZM32.3281 25.2031 L32.3281 57.4688 L9.9062 25.2031 L32.3281 25.2031 Z" + /><glyph unicode="5" horiz-adv-x="55.615234" d="M4.1562 18.75 L13.375 19.5312 Q14.4062 12.7969 18.1406 9.3984 Q21.875 6 27.1562 6 Q33.5 6 37.8906 10.7891 Q42.2812 15.5781 42.2812 23.4844 Q42.2812 31 38.0625 35.3516 Q33.8438 39.7031 27 39.7031 Q22.75 39.7031 19.3359 37.7734 Q15.9219 35.8438 13.9688 32.7656 L5.7188 33.8438 L12.6406 70.6094 L48.25 70.6094 L48.25 62.2031 L19.6719 62.2031 L15.8281 42.9688 Q22.2656 47.4688 29.3438 47.4688 Q38.7188 47.4688 45.1641 40.9688 Q51.6094 34.4688 51.6094 24.2656 Q51.6094 14.5469 45.9531 7.4688 Q39.0625 -1.2188 27.1562 -1.2188 Q17.3906 -1.2188 11.2109 4.25 Q5.0312 9.7188 4.1562 18.75 Z" + /><glyph unicode="6" horiz-adv-x="55.615234" d="M49.75 54.0469 L41.0156 53.375 Q39.8438 58.5469 37.7031 60.8906 Q34.125 64.6562 28.9062 64.6562 Q24.7031 64.6562 21.5312 62.3125 Q17.3906 59.2812 14.9922 53.4688 Q12.5938 47.6562 12.5 36.9219 Q15.6719 41.75 20.2656 44.0938 Q24.8594 46.4375 29.8906 46.4375 Q38.6719 46.4375 44.8516 39.9688 Q51.0312 33.5 51.0312 23.25 Q51.0312 16.5 48.125 10.7188 Q45.2188 4.9375 40.1406 1.8594 Q35.0625 -1.2188 28.6094 -1.2188 Q17.625 -1.2188 10.6953 6.8594 Q3.7656 14.9375 3.7656 33.5 Q3.7656 54.25 11.4219 63.6719 Q18.1094 71.875 29.4375 71.875 Q37.8906 71.875 43.2891 67.1406 Q48.6875 62.4062 49.75 54.0469 ZM13.875 23.1875 Q13.875 18.6562 15.7969 14.5078 Q17.7188 10.3594 21.1875 8.1797 Q24.6562 6 28.4688 6 Q34.0312 6 38.0391 10.4922 Q42.0469 14.9844 42.0469 22.7031 Q42.0469 30.125 38.0859 34.3984 Q34.125 38.6719 28.125 38.6719 Q22.1719 38.6719 18.0234 34.3984 Q13.875 30.125 13.875 23.1875 Z" + /><glyph unicode="7" horiz-adv-x="55.615234" d="M4.7344 62.2031 L4.7344 70.6562 L51.0781 70.6562 L51.0781 63.8125 Q44.2344 56.5469 37.5234 44.4844 Q30.8125 32.4219 27.1562 19.6719 Q24.5156 10.6875 23.7812 0 L14.75 0 Q14.8906 8.4531 18.0625 20.4141 Q21.2344 32.375 27.1719 43.4844 Q33.1094 54.5938 39.7969 62.2031 L4.7344 62.2031 Z" + /><glyph unicode="8" horiz-adv-x="55.615234" d="M17.6719 38.8125 Q12.2031 40.8281 9.5703 44.5391 Q6.9375 48.25 6.9375 53.4219 Q6.9375 61.2344 12.5547 66.5547 Q18.1719 71.875 27.4844 71.875 Q36.8594 71.875 42.5781 66.4297 Q48.2969 60.9844 48.2969 53.1719 Q48.2969 48.1875 45.6797 44.5078 Q43.0625 40.8281 37.75 38.8125 Q44.3438 36.6719 47.7812 31.8828 Q51.2188 27.0938 51.2188 20.4531 Q51.2188 11.2812 44.7266 5.0312 Q38.2344 -1.2188 27.6406 -1.2188 Q17.0469 -1.2188 10.5469 5.0547 Q4.0469 11.3281 4.0469 20.7031 Q4.0469 27.6875 7.5938 32.3984 Q11.1406 37.1094 17.6719 38.8125 ZM15.9219 53.7188 Q15.9219 48.6406 19.1953 45.4141 Q22.4688 42.1875 27.6875 42.1875 Q32.7656 42.1875 36.0156 45.3828 Q39.2656 48.5781 39.2656 53.2188 Q39.2656 58.0625 35.9141 61.3594 Q32.5625 64.6562 27.5938 64.6562 Q22.5625 64.6562 19.2422 61.4297 Q15.9219 58.2031 15.9219 53.7188 ZM13.0938 20.6562 Q13.0938 16.8906 14.875 13.375 Q16.6562 9.8594 20.1719 7.9297 Q23.6875 6 27.7344 6 Q34.0312 6 38.1328 10.0547 Q42.2344 14.1094 42.2344 20.3594 Q42.2344 26.7031 38.0156 30.8594 Q33.7969 35.0156 27.4375 35.0156 Q21.2344 35.0156 17.1641 30.9141 Q13.0938 26.8125 13.0938 20.6562 Z" + /><glyph unicode="9" horiz-adv-x="55.615234" d="M5.4688 16.5469 L13.9219 17.3281 Q14.9844 11.375 18.0156 8.6875 Q21.0469 6 25.7812 6 Q29.8281 6 32.8828 7.8594 Q35.9375 9.7188 37.8906 12.8203 Q39.8438 15.9219 41.1641 21.1953 Q42.4844 26.4688 42.4844 31.9375 Q42.4844 32.5156 42.4375 33.6875 Q39.7969 29.5 35.2344 26.8828 Q30.6719 24.2656 25.3438 24.2656 Q16.4531 24.2656 10.3047 30.7109 Q4.1562 37.1562 4.1562 47.7031 Q4.1562 58.5938 10.5781 65.2344 Q17 71.875 26.6562 71.875 Q33.6406 71.875 39.4297 68.1172 Q45.2188 64.3594 48.2188 57.3984 Q51.2188 50.4375 51.2188 37.25 Q51.2188 23.5312 48.2422 15.4062 Q45.2656 7.2812 39.3828 3.0312 Q33.5 -1.2188 25.5938 -1.2188 Q17.1875 -1.2188 11.8672 3.4453 Q6.5469 8.1094 5.4688 16.5469 ZM41.4531 48.1406 Q41.4531 55.7188 37.4297 60.1562 Q33.4062 64.5938 27.7344 64.5938 Q21.875 64.5938 17.5312 59.8125 Q13.1875 55.0312 13.1875 47.4062 Q13.1875 40.5781 17.3125 36.3047 Q21.4375 32.0312 27.4844 32.0312 Q33.5938 32.0312 37.5234 36.3047 Q41.4531 40.5781 41.4531 48.1406 Z" + /></font + ><font horiz-adv-x="75.0" id="font2" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:bold;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="t" horiz-adv-x="33.30078" d="M30.9531 51.8594 L30.9531 40.9219 L21.5781 40.9219 L21.5781 20.0156 Q21.5781 13.6719 21.8516 12.625 Q22.125 11.5781 23.0781 10.8906 Q24.0312 10.2031 25.3906 10.2031 Q27.2969 10.2031 30.9062 11.5312 L32.0781 0.875 Q27.2969 -1.1719 21.2344 -1.1719 Q17.5312 -1.1719 14.5547 0.0703 Q11.5781 1.3125 10.1875 3.2969 Q8.7969 5.2812 8.25 8.6406 Q7.8125 11.0312 7.8125 18.3125 L7.8125 40.9219 L1.5156 40.9219 L1.5156 51.8594 L7.8125 51.8594 L7.8125 62.1562 L21.5781 70.1719 L21.5781 51.8594 L30.9531 51.8594 Z" + /><glyph unicode="r" horiz-adv-x="38.916016" d="M20.3125 0 L6.5938 0 L6.5938 51.8594 L19.3438 51.8594 L19.3438 44.4844 Q22.6094 49.7031 25.2188 51.3672 Q27.8281 53.0312 31.1562 53.0312 Q35.8438 53.0312 40.1875 50.4375 L35.9375 38.4844 Q32.4688 40.7188 29.5 40.7188 Q26.6094 40.7188 24.6094 39.1328 Q22.6094 37.5469 21.4609 33.3984 Q20.3125 29.25 20.3125 16.0156 L20.3125 0 Z" + /><glyph unicode="e" horiz-adv-x="55.615234" d="M37.2031 16.5 L50.875 14.2031 Q48.25 6.6875 42.5547 2.7578 Q36.8594 -1.1719 28.3281 -1.1719 Q14.7969 -1.1719 8.2969 7.6719 Q3.1719 14.75 3.1719 25.5312 Q3.1719 38.4219 9.9141 45.7266 Q16.6562 53.0312 26.9531 53.0312 Q38.5312 53.0312 45.2188 45.3906 Q51.9062 37.75 51.6094 21.9688 L17.2344 21.9688 Q17.3906 15.875 20.5625 12.4766 Q23.7344 9.0781 28.4688 9.0781 Q31.6875 9.0781 33.8828 10.8359 Q36.0781 12.5938 37.2031 16.5 ZM37.9844 30.375 Q37.8438 36.3281 34.9141 39.4297 Q31.9844 42.5312 27.7812 42.5312 Q23.2969 42.5312 20.3594 39.2656 Q17.4375 35.9844 17.4844 30.375 L37.9844 30.375 Z" + /><glyph unicode="i" horiz-adv-x="27.783203" d="M7.1719 58.8906 L7.1719 71.5781 L20.9062 71.5781 L20.9062 58.8906 L7.1719 58.8906 ZM7.1719 0 L7.1719 51.8594 L20.9062 51.8594 L20.9062 0 L7.1719 0 Z" + /><glyph unicode="d" horiz-adv-x="61.083984" d="M54.7344 0 L42 0 L42 7.625 Q38.8125 3.1719 34.4922 1 Q30.1719 -1.1719 25.7812 -1.1719 Q16.8438 -1.1719 10.4766 6.0312 Q4.1094 13.2344 4.1094 26.125 Q4.1094 39.3125 10.3047 46.1719 Q16.5 53.0312 25.9844 53.0312 Q34.6719 53.0312 41.0156 45.7969 L41.0156 71.5781 L54.7344 71.5781 L54.7344 0 ZM18.1094 27.0469 Q18.1094 18.75 20.4062 15.0469 Q23.7344 9.6719 29.6875 9.6719 Q34.4219 9.6719 37.7422 13.6953 Q41.0625 17.7188 41.0625 25.7344 Q41.0625 34.6719 37.8438 38.6016 Q34.625 42.5312 29.5938 42.5312 Q24.7031 42.5312 21.4062 38.6484 Q18.1094 34.7656 18.1094 27.0469 Z" + /><glyph unicode="o" horiz-adv-x="61.083984" d="M4 26.6562 Q4 33.5 7.375 39.8984 Q10.75 46.2969 16.9219 49.6641 Q23.0938 53.0312 30.7188 53.0312 Q42.4844 53.0312 50 45.3906 Q57.5156 37.75 57.5156 26.0781 Q57.5156 14.3125 49.9219 6.5703 Q42.3281 -1.1719 30.8125 -1.1719 Q23.6875 -1.1719 17.2188 2.0547 Q10.75 5.2812 7.375 11.5 Q4 17.7188 4 26.6562 ZM18.0625 25.9219 Q18.0625 18.2188 21.7266 14.1172 Q25.3906 10.0156 30.7656 10.0156 Q36.1406 10.0156 39.7734 14.1172 Q43.4062 18.2188 43.4062 26.0312 Q43.4062 33.6406 39.7734 37.7422 Q36.1406 41.8438 30.7656 41.8438 Q25.3906 41.8438 21.7266 37.7422 Q18.0625 33.6406 18.0625 25.9219 Z" + /><glyph unicode="C" horiz-adv-x="72.2168" d="M53.0781 26.3125 L67.0938 21.875 Q63.875 10.1562 56.375 4.4688 Q48.875 -1.2188 37.3594 -1.2188 Q23.0938 -1.2188 13.9141 8.5234 Q4.7344 18.2656 4.7344 35.1562 Q4.7344 53.0312 13.9609 62.9141 Q23.1875 72.7969 38.2344 72.7969 Q51.375 72.7969 59.5781 65.0469 Q64.4531 60.4531 66.8906 51.8594 L52.5938 48.4375 Q51.3125 54 47.2891 57.2266 Q43.2656 60.4531 37.5 60.4531 Q29.5469 60.4531 24.5859 54.7422 Q19.625 49.0312 19.625 36.2344 Q19.625 22.6562 24.5078 16.8984 Q29.3906 11.1406 37.2031 11.1406 Q42.9688 11.1406 47.1172 14.7969 Q51.2656 18.4531 53.0781 26.3125 Z" + /></font + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >10</text + ></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >20</text + ></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >30</text + ></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >40</text + ></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >50</text + ></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >60</text + ></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >70</text + ></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >80</text + ></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >90</text + ></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="325" style="fill:none;" x1="73" x2="77.34" y1="325" + /><line y2="276" style="fill:none;" x1="73" x2="77.34" y1="276" + /><line y2="227" style="fill:none;" x1="73" x2="77.34" y1="227" + /><line y2="178" style="fill:none;" x1="73" x2="77.34" y1="178" + /><line y2="129" style="fill:none;" x1="73" x2="77.34" y1="129" + /><line y2="80" style="fill:none;" x1="73" x2="77.34" y1="80" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="325" style="fill:none;" x1="507" x2="502.66" y1="325" + /><line y2="276" style="fill:none;" x1="507" x2="502.66" y1="276" + /><line y2="227" style="fill:none;" x1="507" x2="502.66" y1="227" + /><line y2="178" style="fill:none;" x1="507" x2="502.66" y1="178" + /><line y2="129" style="fill:none;" x1="507" x2="502.66" y1="129" + /><line y2="80" style="fill:none;" x1="507" x2="502.66" y1="80" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >0</text + ></g + ><g transform="translate(67.6667,325)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-15" xml:space="preserve" y="5.5" style="stroke:none;" + >50</text + ></g + ><g transform="translate(67.6667,276)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >100</text + ></g + ><g transform="translate(67.6667,227)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >150</text + ></g + ><g transform="translate(67.6667,178)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >200</text + ></g + ><g transform="translate(67.6667,129)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >250</text + ></g + ><g transform="translate(67.6667,80)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >300</text + ></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >350</text + ></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><text x="-26" xml:space="preserve" y="-4" style="stroke:none;" + >Codiert</text + ></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 75.1 L77.5684 261.7875 L82.1368 289.4718 L86.7053 363.9599 L91.2737 307.2254 L95.8421 362.0164 L100.4105 370.8774 L104.979 354.3807 L109.5474 361.9688 L114.1158 362.5128 L118.6842 349.6222 L123.2526 358.0281 L127.8211 351.2132 L132.3895 357.7471 L136.9579 358.6982 L141.5263 341.2717 L146.0947 353.6311 L150.6632 355.0342 L155.2316 357.9427 L159.8 353.4687 L164.3684 357.2768 L168.9368 362.5093 L173.5053 343.5215 L178.0737 345.155 L182.6421 347.9055 L187.2105 356.3438 L191.7789 349.7505 L196.3474 366.6401 L200.9158 370.0116 L205.4842 347.3291 L210.0526 357.1107 L214.621 350.9171 L219.1895 352.1744 L223.7579 359.4998 L228.3263 348.958 L232.8947 341.2498 L237.4632 362.6831 L242.0316 353.3522 L246.6 345.781 L251.1684 361.8681 L255.7368 357.3319 L260.3053 354.6476 L264.8737 355.92 L269.4421 362.0248 L274.0105 366.566 L278.5789 368.0974 L283.1474 359.7258 L287.7158 370.2862 L292.2842 353.42 L296.8526 370.2862 L301.4211 359.7258 L305.9895 368.0974 L310.5579 366.566 L315.1263 362.0248 L319.6947 355.92 L324.2632 354.6476 L328.8316 357.3319 L333.4 361.8681 L337.9684 345.781 L342.5368 353.3522 L347.1053 362.6831 L351.6737 341.2498 L356.2421 348.958 L360.8105 359.4998 L365.3789 352.1744 L369.9474 350.9171 L374.5158 357.1107 L379.0842 347.3291 L383.6526 370.0116 L388.221 366.6401 L392.7895 349.7505 L397.3579 356.3438 L401.9263 347.9055 L406.4948 345.155 L411.0632 343.5215 L415.6316 362.5093 L420.2 357.2768 L424.7684 353.4687 L429.3369 357.9427 L433.9053 355.0342 L438.4737 353.6311 L443.0421 341.2717 L447.6105 358.6982 L452.179 357.7471 L456.7474 351.2132 L461.3158 358.0281 L465.8842 349.6222 L470.4526 362.5128 L475.0211 361.9688 L479.5895 354.3807 L484.1579 370.8774 L488.7263 362.0164 L493.2947 307.2254 L497.8632 363.9599 L502.4316 289.4718 L507 261.7875" style="fill:none; fill-rule:evenodd;" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/fig3.pdf b/buch/papers/reedsolomon/RS presentation/images/fig3.pdf Binary files differnew file mode 100644 index 0000000..7d29d1e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig3.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig3.svg b/buch/papers/reedsolomon/RS presentation/images/fig3.svg new file mode 100644 index 0000000..e0c7072 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig3.svg @@ -0,0 +1,180 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ><font horiz-adv-x="75.0" id="font1" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:normal;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="0" horiz-adv-x="55.615234" d="M4.1562 35.2969 Q4.1562 48 6.7656 55.7422 Q9.375 63.4844 14.5234 67.6797 Q19.6719 71.875 27.4844 71.875 Q33.25 71.875 37.5938 69.5547 Q41.9375 67.2344 44.7734 62.8672 Q47.6094 58.5 49.2188 52.2266 Q50.8281 45.9531 50.8281 35.2969 Q50.8281 22.7031 48.2422 14.9688 Q45.6562 7.2344 40.5078 3.0078 Q35.3594 -1.2188 27.4844 -1.2188 Q17.1406 -1.2188 11.2344 6.2031 Q4.1562 15.1406 4.1562 35.2969 ZM13.1875 35.2969 Q13.1875 17.6719 17.3125 11.8359 Q21.4375 6 27.4844 6 Q33.5469 6 37.6719 11.8594 Q41.7969 17.7188 41.7969 35.2969 Q41.7969 52.9844 37.6719 58.7891 Q33.5469 64.5938 27.3906 64.5938 Q21.3438 64.5938 17.7188 59.4688 Q13.1875 52.9375 13.1875 35.2969 Z" + /><glyph unicode="1" horiz-adv-x="55.615234" d="M37.25 0 L28.4688 0 L28.4688 56 Q25.2969 52.9844 20.1406 49.9531 Q14.9844 46.9219 10.8906 45.4062 L10.8906 53.9062 Q18.2656 57.375 23.7812 62.3047 Q29.2969 67.2344 31.5938 71.875 L37.25 71.875 L37.25 0 Z" + /><glyph unicode="2" horiz-adv-x="55.615234" d="M50.3438 8.4531 L50.3438 0 L3.0312 0 Q2.9375 3.1719 4.0469 6.1094 Q5.8594 10.9375 9.8359 15.625 Q13.8125 20.3125 21.3438 26.4688 Q33.0156 36.0312 37.1172 41.625 Q41.2188 47.2188 41.2188 52.2031 Q41.2188 57.4219 37.4766 61.0078 Q33.7344 64.5938 27.7344 64.5938 Q21.3906 64.5938 17.5781 60.7891 Q13.7656 56.9844 13.7188 50.25 L4.6875 51.1719 Q5.6094 61.2812 11.6641 66.5781 Q17.7188 71.875 27.9375 71.875 Q38.2344 71.875 44.2422 66.1641 Q50.25 60.4531 50.25 52 Q50.25 47.7031 48.4922 43.5547 Q46.7344 39.4062 42.6562 34.8125 Q38.5781 30.2188 29.1094 22.2188 Q21.1875 15.5781 18.9453 13.2109 Q16.7031 10.8438 15.2344 8.4531 L50.3438 8.4531 Z" + /><glyph unicode="3" horiz-adv-x="55.615234" d="M4.2031 18.8906 L12.9844 20.0625 Q14.5 12.5938 18.1406 9.2969 Q21.7812 6 27 6 Q33.2031 6 37.4766 10.2969 Q41.75 14.5938 41.75 20.9531 Q41.75 27 37.7969 30.9297 Q33.8438 34.8594 27.7344 34.8594 Q25.25 34.8594 21.5312 33.8906 L22.5156 41.6094 Q23.3906 41.5 23.9219 41.5 Q29.5469 41.5 34.0391 44.4297 Q38.5312 47.3594 38.5312 53.4688 Q38.5312 58.2969 35.2578 61.4766 Q31.9844 64.6562 26.8125 64.6562 Q21.6875 64.6562 18.2656 61.4297 Q14.8438 58.2031 13.875 51.7656 L5.0781 53.3281 Q6.6875 62.1562 12.3984 67.0156 Q18.1094 71.875 26.6094 71.875 Q32.4688 71.875 37.3984 69.3594 Q42.3281 66.8438 44.9453 62.5 Q47.5625 58.1562 47.5625 53.2656 Q47.5625 48.6406 45.0703 44.8281 Q42.5781 41.0156 37.7031 38.7656 Q44.0469 37.3125 47.5625 32.6953 Q51.0781 28.0781 51.0781 21.1406 Q51.0781 11.7656 44.2422 5.25 Q37.4062 -1.2656 26.9531 -1.2656 Q17.5312 -1.2656 11.3047 4.3516 Q5.0781 9.9688 4.2031 18.8906 Z" + /><glyph unicode="4" horiz-adv-x="55.615234" d="M32.3281 0 L32.3281 17.1406 L1.2656 17.1406 L1.2656 25.2031 L33.9375 71.5781 L41.1094 71.5781 L41.1094 25.2031 L50.7812 25.2031 L50.7812 17.1406 L41.1094 17.1406 L41.1094 0 L32.3281 0 ZM32.3281 25.2031 L32.3281 57.4688 L9.9062 25.2031 L32.3281 25.2031 Z" + /><glyph unicode="5" horiz-adv-x="55.615234" d="M4.1562 18.75 L13.375 19.5312 Q14.4062 12.7969 18.1406 9.3984 Q21.875 6 27.1562 6 Q33.5 6 37.8906 10.7891 Q42.2812 15.5781 42.2812 23.4844 Q42.2812 31 38.0625 35.3516 Q33.8438 39.7031 27 39.7031 Q22.75 39.7031 19.3359 37.7734 Q15.9219 35.8438 13.9688 32.7656 L5.7188 33.8438 L12.6406 70.6094 L48.25 70.6094 L48.25 62.2031 L19.6719 62.2031 L15.8281 42.9688 Q22.2656 47.4688 29.3438 47.4688 Q38.7188 47.4688 45.1641 40.9688 Q51.6094 34.4688 51.6094 24.2656 Q51.6094 14.5469 45.9531 7.4688 Q39.0625 -1.2188 27.1562 -1.2188 Q17.3906 -1.2188 11.2109 4.25 Q5.0312 9.7188 4.1562 18.75 Z" + /><glyph unicode="6" horiz-adv-x="55.615234" d="M49.75 54.0469 L41.0156 53.375 Q39.8438 58.5469 37.7031 60.8906 Q34.125 64.6562 28.9062 64.6562 Q24.7031 64.6562 21.5312 62.3125 Q17.3906 59.2812 14.9922 53.4688 Q12.5938 47.6562 12.5 36.9219 Q15.6719 41.75 20.2656 44.0938 Q24.8594 46.4375 29.8906 46.4375 Q38.6719 46.4375 44.8516 39.9688 Q51.0312 33.5 51.0312 23.25 Q51.0312 16.5 48.125 10.7188 Q45.2188 4.9375 40.1406 1.8594 Q35.0625 -1.2188 28.6094 -1.2188 Q17.625 -1.2188 10.6953 6.8594 Q3.7656 14.9375 3.7656 33.5 Q3.7656 54.25 11.4219 63.6719 Q18.1094 71.875 29.4375 71.875 Q37.8906 71.875 43.2891 67.1406 Q48.6875 62.4062 49.75 54.0469 ZM13.875 23.1875 Q13.875 18.6562 15.7969 14.5078 Q17.7188 10.3594 21.1875 8.1797 Q24.6562 6 28.4688 6 Q34.0312 6 38.0391 10.4922 Q42.0469 14.9844 42.0469 22.7031 Q42.0469 30.125 38.0859 34.3984 Q34.125 38.6719 28.125 38.6719 Q22.1719 38.6719 18.0234 34.3984 Q13.875 30.125 13.875 23.1875 Z" + /><glyph unicode="7" horiz-adv-x="55.615234" d="M4.7344 62.2031 L4.7344 70.6562 L51.0781 70.6562 L51.0781 63.8125 Q44.2344 56.5469 37.5234 44.4844 Q30.8125 32.4219 27.1562 19.6719 Q24.5156 10.6875 23.7812 0 L14.75 0 Q14.8906 8.4531 18.0625 20.4141 Q21.2344 32.375 27.1719 43.4844 Q33.1094 54.5938 39.7969 62.2031 L4.7344 62.2031 Z" + /><glyph unicode="8" horiz-adv-x="55.615234" d="M17.6719 38.8125 Q12.2031 40.8281 9.5703 44.5391 Q6.9375 48.25 6.9375 53.4219 Q6.9375 61.2344 12.5547 66.5547 Q18.1719 71.875 27.4844 71.875 Q36.8594 71.875 42.5781 66.4297 Q48.2969 60.9844 48.2969 53.1719 Q48.2969 48.1875 45.6797 44.5078 Q43.0625 40.8281 37.75 38.8125 Q44.3438 36.6719 47.7812 31.8828 Q51.2188 27.0938 51.2188 20.4531 Q51.2188 11.2812 44.7266 5.0312 Q38.2344 -1.2188 27.6406 -1.2188 Q17.0469 -1.2188 10.5469 5.0547 Q4.0469 11.3281 4.0469 20.7031 Q4.0469 27.6875 7.5938 32.3984 Q11.1406 37.1094 17.6719 38.8125 ZM15.9219 53.7188 Q15.9219 48.6406 19.1953 45.4141 Q22.4688 42.1875 27.6875 42.1875 Q32.7656 42.1875 36.0156 45.3828 Q39.2656 48.5781 39.2656 53.2188 Q39.2656 58.0625 35.9141 61.3594 Q32.5625 64.6562 27.5938 64.6562 Q22.5625 64.6562 19.2422 61.4297 Q15.9219 58.2031 15.9219 53.7188 ZM13.0938 20.6562 Q13.0938 16.8906 14.875 13.375 Q16.6562 9.8594 20.1719 7.9297 Q23.6875 6 27.7344 6 Q34.0312 6 38.1328 10.0547 Q42.2344 14.1094 42.2344 20.3594 Q42.2344 26.7031 38.0156 30.8594 Q33.7969 35.0156 27.4375 35.0156 Q21.2344 35.0156 17.1641 30.9141 Q13.0938 26.8125 13.0938 20.6562 Z" + /><glyph unicode="9" horiz-adv-x="55.615234" d="M5.4688 16.5469 L13.9219 17.3281 Q14.9844 11.375 18.0156 8.6875 Q21.0469 6 25.7812 6 Q29.8281 6 32.8828 7.8594 Q35.9375 9.7188 37.8906 12.8203 Q39.8438 15.9219 41.1641 21.1953 Q42.4844 26.4688 42.4844 31.9375 Q42.4844 32.5156 42.4375 33.6875 Q39.7969 29.5 35.2344 26.8828 Q30.6719 24.2656 25.3438 24.2656 Q16.4531 24.2656 10.3047 30.7109 Q4.1562 37.1562 4.1562 47.7031 Q4.1562 58.5938 10.5781 65.2344 Q17 71.875 26.6562 71.875 Q33.6406 71.875 39.4297 68.1172 Q45.2188 64.3594 48.2188 57.3984 Q51.2188 50.4375 51.2188 37.25 Q51.2188 23.5312 48.2422 15.4062 Q45.2656 7.2812 39.3828 3.0312 Q33.5 -1.2188 25.5938 -1.2188 Q17.1875 -1.2188 11.8672 3.4453 Q6.5469 8.1094 5.4688 16.5469 ZM41.4531 48.1406 Q41.4531 55.7188 37.4297 60.1562 Q33.4062 64.5938 27.7344 64.5938 Q21.875 64.5938 17.5312 59.8125 Q13.1875 55.0312 13.1875 47.4062 Q13.1875 40.5781 17.3125 36.3047 Q21.4375 32.0312 27.4844 32.0312 Q33.5938 32.0312 37.5234 36.3047 Q41.4531 40.5781 41.4531 48.1406 Z" + /><glyph unicode="." horiz-adv-x="27.783203" d="M9.0781 0 L9.0781 10.0156 L19.0938 10.0156 L19.0938 0 L9.0781 0 Z" + /></font + ><font horiz-adv-x="75.0" id="font2" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:bold;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="r" horiz-adv-x="38.916016" d="M20.3125 0 L6.5938 0 L6.5938 51.8594 L19.3438 51.8594 L19.3438 44.4844 Q22.6094 49.7031 25.2188 51.3672 Q27.8281 53.0312 31.1562 53.0312 Q35.8438 53.0312 40.1875 50.4375 L35.9375 38.4844 Q32.4688 40.7188 29.5 40.7188 Q26.6094 40.7188 24.6094 39.1328 Q22.6094 37.5469 21.4609 33.3984 Q20.3125 29.25 20.3125 16.0156 L20.3125 0 Z" + /><glyph unicode="l" horiz-adv-x="27.783203" d="M7.1719 0 L7.1719 71.5781 L20.9062 71.5781 L20.9062 0 L7.1719 0 Z" + /><glyph unicode="h" horiz-adv-x="61.083984" d="M20.8438 71.5781 L20.8438 45.2656 Q27.4844 53.0312 36.7188 53.0312 Q41.4531 53.0312 45.2656 51.2734 Q49.0781 49.5156 51.0078 46.7812 Q52.9375 44.0469 53.6406 40.7266 Q54.3438 37.4062 54.3438 30.4219 L54.3438 0 L40.625 0 L40.625 27.3906 Q40.625 35.5469 39.8438 37.7422 Q39.0625 39.9375 37.0859 41.2344 Q35.1094 42.5312 32.125 42.5312 Q28.7188 42.5312 26.0312 40.8672 Q23.3438 39.2031 22.0938 35.8594 Q20.8438 32.5156 20.8438 25.9844 L20.8438 0 L7.125 0 L7.125 71.5781 L20.8438 71.5781 Z" + /><glyph unicode="e" horiz-adv-x="55.615234" d="M37.2031 16.5 L50.875 14.2031 Q48.25 6.6875 42.5547 2.7578 Q36.8594 -1.1719 28.3281 -1.1719 Q14.7969 -1.1719 8.2969 7.6719 Q3.1719 14.75 3.1719 25.5312 Q3.1719 38.4219 9.9141 45.7266 Q16.6562 53.0312 26.9531 53.0312 Q38.5312 53.0312 45.2188 45.3906 Q51.9062 37.75 51.6094 21.9688 L17.2344 21.9688 Q17.3906 15.875 20.5625 12.4766 Q23.7344 9.0781 28.4688 9.0781 Q31.6875 9.0781 33.8828 10.8359 Q36.0781 12.5938 37.2031 16.5 ZM37.9844 30.375 Q37.8438 36.3281 34.9141 39.4297 Q31.9844 42.5312 27.7812 42.5312 Q23.2969 42.5312 20.3594 39.2656 Q17.4375 35.9844 17.4844 30.375 L37.9844 30.375 Z" + /><glyph unicode="F" horiz-adv-x="61.083984" d="M7.375 0 L7.375 71.5781 L56.4531 71.5781 L56.4531 59.4688 L21.8281 59.4688 L21.8281 42.5312 L51.7031 42.5312 L51.7031 30.4219 L21.8281 30.4219 L21.8281 0 L7.375 0 Z" + /></font + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >10</text + ></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >20</text + ></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >30</text + ></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >40</text + ></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >50</text + ></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >60</text + ></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >70</text + ></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >80</text + ></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >90</text + ></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="339.7" style="fill:none;" x1="73" x2="77.34" y1="339.7" + /><line y2="305.4" style="fill:none;" x1="73" x2="77.34" y1="305.4" + /><line y2="271.1" style="fill:none;" x1="73" x2="77.34" y1="271.1" + /><line y2="236.8" style="fill:none;" x1="73" x2="77.34" y1="236.8" + /><line y2="202.5" style="fill:none;" x1="73" x2="77.34" y1="202.5" + /><line y2="168.2" style="fill:none;" x1="73" x2="77.34" y1="168.2" + /><line y2="133.9" style="fill:none;" x1="73" x2="77.34" y1="133.9" + /><line y2="99.6" style="fill:none;" x1="73" x2="77.34" y1="99.6" + /><line y2="65.3" style="fill:none;" x1="73" x2="77.34" y1="65.3" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="339.7" style="fill:none;" x1="507" x2="502.66" y1="339.7" + /><line y2="305.4" style="fill:none;" x1="507" x2="502.66" y1="305.4" + /><line y2="271.1" style="fill:none;" x1="507" x2="502.66" y1="271.1" + /><line y2="236.8" style="fill:none;" x1="507" x2="502.66" y1="236.8" + /><line y2="202.5" style="fill:none;" x1="507" x2="502.66" y1="202.5" + /><line y2="168.2" style="fill:none;" x1="507" x2="502.66" y1="168.2" + /><line y2="133.9" style="fill:none;" x1="507" x2="502.66" y1="133.9" + /><line y2="99.6" style="fill:none;" x1="507" x2="502.66" y1="99.6" + /><line y2="65.3" style="fill:none;" x1="507" x2="502.66" y1="65.3" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >0</text + ></g + ><g transform="translate(67.6667,339.7)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.2</text + ></g + ><g transform="translate(67.6667,305.4)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.4</text + ></g + ><g transform="translate(67.6667,271.1)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.6</text + ></g + ><g transform="translate(67.6667,236.8)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.8</text + ></g + ><g transform="translate(67.6667,202.5)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >1</text + ></g + ><g transform="translate(67.6667,168.2)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >1.2</text + ></g + ><g transform="translate(67.6667,133.9)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >1.4</text + ></g + ><g transform="translate(67.6667,99.6)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >1.6</text + ></g + ><g transform="translate(67.6667,65.3)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >1.8</text + ></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >2</text + ></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><text x="-22" xml:space="preserve" y="-4" style="stroke:none;" + >Fehler</text + ></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 374 L77.5684 374 L82.1368 374 L86.7053 374 L91.2737 374 L95.8421 374 L100.4105 31 L104.979 374 L109.5474 374 L114.1158 374 L118.6842 374 L123.2526 374 L127.8211 374 L132.3895 374 L136.9579 374 L141.5263 374 L146.0947 374 L150.6632 374 L155.2316 374 L159.8 374 L164.3684 31 L168.9368 374 L173.5053 374 L178.0737 374 L182.6421 374 L187.2105 374 L191.7789 374 L196.3474 374 L200.9158 374 L205.4842 374 L210.0526 374 L214.621 374 L219.1895 374 L223.7579 374 L228.3263 374 L232.8947 374 L237.4632 374 L242.0316 374 L246.6 374 L251.1684 374 L255.7368 374 L260.3053 374 L264.8737 374 L269.4421 374 L274.0105 374 L278.5789 374 L283.1474 374 L287.7158 374 L292.2842 374 L296.8526 374 L301.4211 374 L305.9895 374 L310.5579 374 L315.1263 374 L319.6947 374 L324.2632 374 L328.8316 374 L333.4 374 L337.9684 374 L342.5368 374 L347.1053 374 L351.6737 374 L356.2421 374 L360.8105 374 L365.3789 374 L369.9474 374 L374.5158 374 L379.0842 374 L383.6526 374 L388.221 374 L392.7895 374 L397.3579 374 L401.9263 374 L406.4948 374 L411.0632 202.5 L415.6316 374 L420.2 374 L424.7684 374 L429.3369 374 L433.9053 374 L438.4737 374 L443.0421 374 L447.6105 374 L452.179 374 L456.7474 374 L461.3158 374 L465.8842 374 L470.4526 374 L475.0211 374 L479.5895 374 L484.1579 374 L488.7263 374 L493.2947 374 L497.8632 374 L502.4316 374 L507 374" style="fill:none; fill-rule:evenodd;" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/fig4.pdf b/buch/papers/reedsolomon/RS presentation/images/fig4.pdf Binary files differnew file mode 100644 index 0000000..394a673 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig4.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig4.svg b/buch/papers/reedsolomon/RS presentation/images/fig4.svg new file mode 100644 index 0000000..4bf2864 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig4.svg @@ -0,0 +1,164 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ><font horiz-adv-x="75.0" id="font1" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:normal;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="0" horiz-adv-x="55.615234" d="M4.1562 35.2969 Q4.1562 48 6.7656 55.7422 Q9.375 63.4844 14.5234 67.6797 Q19.6719 71.875 27.4844 71.875 Q33.25 71.875 37.5938 69.5547 Q41.9375 67.2344 44.7734 62.8672 Q47.6094 58.5 49.2188 52.2266 Q50.8281 45.9531 50.8281 35.2969 Q50.8281 22.7031 48.2422 14.9688 Q45.6562 7.2344 40.5078 3.0078 Q35.3594 -1.2188 27.4844 -1.2188 Q17.1406 -1.2188 11.2344 6.2031 Q4.1562 15.1406 4.1562 35.2969 ZM13.1875 35.2969 Q13.1875 17.6719 17.3125 11.8359 Q21.4375 6 27.4844 6 Q33.5469 6 37.6719 11.8594 Q41.7969 17.7188 41.7969 35.2969 Q41.7969 52.9844 37.6719 58.7891 Q33.5469 64.5938 27.3906 64.5938 Q21.3438 64.5938 17.7188 59.4688 Q13.1875 52.9375 13.1875 35.2969 Z" + /><glyph unicode="1" horiz-adv-x="55.615234" d="M37.25 0 L28.4688 0 L28.4688 56 Q25.2969 52.9844 20.1406 49.9531 Q14.9844 46.9219 10.8906 45.4062 L10.8906 53.9062 Q18.2656 57.375 23.7812 62.3047 Q29.2969 67.2344 31.5938 71.875 L37.25 71.875 L37.25 0 Z" + /><glyph unicode="2" horiz-adv-x="55.615234" d="M50.3438 8.4531 L50.3438 0 L3.0312 0 Q2.9375 3.1719 4.0469 6.1094 Q5.8594 10.9375 9.8359 15.625 Q13.8125 20.3125 21.3438 26.4688 Q33.0156 36.0312 37.1172 41.625 Q41.2188 47.2188 41.2188 52.2031 Q41.2188 57.4219 37.4766 61.0078 Q33.7344 64.5938 27.7344 64.5938 Q21.3906 64.5938 17.5781 60.7891 Q13.7656 56.9844 13.7188 50.25 L4.6875 51.1719 Q5.6094 61.2812 11.6641 66.5781 Q17.7188 71.875 27.9375 71.875 Q38.2344 71.875 44.2422 66.1641 Q50.25 60.4531 50.25 52 Q50.25 47.7031 48.4922 43.5547 Q46.7344 39.4062 42.6562 34.8125 Q38.5781 30.2188 29.1094 22.2188 Q21.1875 15.5781 18.9453 13.2109 Q16.7031 10.8438 15.2344 8.4531 L50.3438 8.4531 Z" + /><glyph unicode="3" horiz-adv-x="55.615234" d="M4.2031 18.8906 L12.9844 20.0625 Q14.5 12.5938 18.1406 9.2969 Q21.7812 6 27 6 Q33.2031 6 37.4766 10.2969 Q41.75 14.5938 41.75 20.9531 Q41.75 27 37.7969 30.9297 Q33.8438 34.8594 27.7344 34.8594 Q25.25 34.8594 21.5312 33.8906 L22.5156 41.6094 Q23.3906 41.5 23.9219 41.5 Q29.5469 41.5 34.0391 44.4297 Q38.5312 47.3594 38.5312 53.4688 Q38.5312 58.2969 35.2578 61.4766 Q31.9844 64.6562 26.8125 64.6562 Q21.6875 64.6562 18.2656 61.4297 Q14.8438 58.2031 13.875 51.7656 L5.0781 53.3281 Q6.6875 62.1562 12.3984 67.0156 Q18.1094 71.875 26.6094 71.875 Q32.4688 71.875 37.3984 69.3594 Q42.3281 66.8438 44.9453 62.5 Q47.5625 58.1562 47.5625 53.2656 Q47.5625 48.6406 45.0703 44.8281 Q42.5781 41.0156 37.7031 38.7656 Q44.0469 37.3125 47.5625 32.6953 Q51.0781 28.0781 51.0781 21.1406 Q51.0781 11.7656 44.2422 5.25 Q37.4062 -1.2656 26.9531 -1.2656 Q17.5312 -1.2656 11.3047 4.3516 Q5.0781 9.9688 4.2031 18.8906 Z" + /><glyph unicode="4" horiz-adv-x="55.615234" d="M32.3281 0 L32.3281 17.1406 L1.2656 17.1406 L1.2656 25.2031 L33.9375 71.5781 L41.1094 71.5781 L41.1094 25.2031 L50.7812 25.2031 L50.7812 17.1406 L41.1094 17.1406 L41.1094 0 L32.3281 0 ZM32.3281 25.2031 L32.3281 57.4688 L9.9062 25.2031 L32.3281 25.2031 Z" + /><glyph unicode="5" horiz-adv-x="55.615234" d="M4.1562 18.75 L13.375 19.5312 Q14.4062 12.7969 18.1406 9.3984 Q21.875 6 27.1562 6 Q33.5 6 37.8906 10.7891 Q42.2812 15.5781 42.2812 23.4844 Q42.2812 31 38.0625 35.3516 Q33.8438 39.7031 27 39.7031 Q22.75 39.7031 19.3359 37.7734 Q15.9219 35.8438 13.9688 32.7656 L5.7188 33.8438 L12.6406 70.6094 L48.25 70.6094 L48.25 62.2031 L19.6719 62.2031 L15.8281 42.9688 Q22.2656 47.4688 29.3438 47.4688 Q38.7188 47.4688 45.1641 40.9688 Q51.6094 34.4688 51.6094 24.2656 Q51.6094 14.5469 45.9531 7.4688 Q39.0625 -1.2188 27.1562 -1.2188 Q17.3906 -1.2188 11.2109 4.25 Q5.0312 9.7188 4.1562 18.75 Z" + /><glyph unicode="6" horiz-adv-x="55.615234" d="M49.75 54.0469 L41.0156 53.375 Q39.8438 58.5469 37.7031 60.8906 Q34.125 64.6562 28.9062 64.6562 Q24.7031 64.6562 21.5312 62.3125 Q17.3906 59.2812 14.9922 53.4688 Q12.5938 47.6562 12.5 36.9219 Q15.6719 41.75 20.2656 44.0938 Q24.8594 46.4375 29.8906 46.4375 Q38.6719 46.4375 44.8516 39.9688 Q51.0312 33.5 51.0312 23.25 Q51.0312 16.5 48.125 10.7188 Q45.2188 4.9375 40.1406 1.8594 Q35.0625 -1.2188 28.6094 -1.2188 Q17.625 -1.2188 10.6953 6.8594 Q3.7656 14.9375 3.7656 33.5 Q3.7656 54.25 11.4219 63.6719 Q18.1094 71.875 29.4375 71.875 Q37.8906 71.875 43.2891 67.1406 Q48.6875 62.4062 49.75 54.0469 ZM13.875 23.1875 Q13.875 18.6562 15.7969 14.5078 Q17.7188 10.3594 21.1875 8.1797 Q24.6562 6 28.4688 6 Q34.0312 6 38.0391 10.4922 Q42.0469 14.9844 42.0469 22.7031 Q42.0469 30.125 38.0859 34.3984 Q34.125 38.6719 28.125 38.6719 Q22.1719 38.6719 18.0234 34.3984 Q13.875 30.125 13.875 23.1875 Z" + /><glyph unicode="7" horiz-adv-x="55.615234" d="M4.7344 62.2031 L4.7344 70.6562 L51.0781 70.6562 L51.0781 63.8125 Q44.2344 56.5469 37.5234 44.4844 Q30.8125 32.4219 27.1562 19.6719 Q24.5156 10.6875 23.7812 0 L14.75 0 Q14.8906 8.4531 18.0625 20.4141 Q21.2344 32.375 27.1719 43.4844 Q33.1094 54.5938 39.7969 62.2031 L4.7344 62.2031 Z" + /><glyph unicode="8" horiz-adv-x="55.615234" d="M17.6719 38.8125 Q12.2031 40.8281 9.5703 44.5391 Q6.9375 48.25 6.9375 53.4219 Q6.9375 61.2344 12.5547 66.5547 Q18.1719 71.875 27.4844 71.875 Q36.8594 71.875 42.5781 66.4297 Q48.2969 60.9844 48.2969 53.1719 Q48.2969 48.1875 45.6797 44.5078 Q43.0625 40.8281 37.75 38.8125 Q44.3438 36.6719 47.7812 31.8828 Q51.2188 27.0938 51.2188 20.4531 Q51.2188 11.2812 44.7266 5.0312 Q38.2344 -1.2188 27.6406 -1.2188 Q17.0469 -1.2188 10.5469 5.0547 Q4.0469 11.3281 4.0469 20.7031 Q4.0469 27.6875 7.5938 32.3984 Q11.1406 37.1094 17.6719 38.8125 ZM15.9219 53.7188 Q15.9219 48.6406 19.1953 45.4141 Q22.4688 42.1875 27.6875 42.1875 Q32.7656 42.1875 36.0156 45.3828 Q39.2656 48.5781 39.2656 53.2188 Q39.2656 58.0625 35.9141 61.3594 Q32.5625 64.6562 27.5938 64.6562 Q22.5625 64.6562 19.2422 61.4297 Q15.9219 58.2031 15.9219 53.7188 ZM13.0938 20.6562 Q13.0938 16.8906 14.875 13.375 Q16.6562 9.8594 20.1719 7.9297 Q23.6875 6 27.7344 6 Q34.0312 6 38.1328 10.0547 Q42.2344 14.1094 42.2344 20.3594 Q42.2344 26.7031 38.0156 30.8594 Q33.7969 35.0156 27.4375 35.0156 Q21.2344 35.0156 17.1641 30.9141 Q13.0938 26.8125 13.0938 20.6562 Z" + /><glyph unicode="9" horiz-adv-x="55.615234" d="M5.4688 16.5469 L13.9219 17.3281 Q14.9844 11.375 18.0156 8.6875 Q21.0469 6 25.7812 6 Q29.8281 6 32.8828 7.8594 Q35.9375 9.7188 37.8906 12.8203 Q39.8438 15.9219 41.1641 21.1953 Q42.4844 26.4688 42.4844 31.9375 Q42.4844 32.5156 42.4375 33.6875 Q39.7969 29.5 35.2344 26.8828 Q30.6719 24.2656 25.3438 24.2656 Q16.4531 24.2656 10.3047 30.7109 Q4.1562 37.1562 4.1562 47.7031 Q4.1562 58.5938 10.5781 65.2344 Q17 71.875 26.6562 71.875 Q33.6406 71.875 39.4297 68.1172 Q45.2188 64.3594 48.2188 57.3984 Q51.2188 50.4375 51.2188 37.25 Q51.2188 23.5312 48.2422 15.4062 Q45.2656 7.2812 39.3828 3.0312 Q33.5 -1.2188 25.5938 -1.2188 Q17.1875 -1.2188 11.8672 3.4453 Q6.5469 8.1094 5.4688 16.5469 ZM41.4531 48.1406 Q41.4531 55.7188 37.4297 60.1562 Q33.4062 64.5938 27.7344 64.5938 Q21.875 64.5938 17.5312 59.8125 Q13.1875 55.0312 13.1875 47.4062 Q13.1875 40.5781 17.3125 36.3047 Q21.4375 32.0312 27.4844 32.0312 Q33.5938 32.0312 37.5234 36.3047 Q41.4531 40.5781 41.4531 48.1406 Z" + /></font + ><font horiz-adv-x="75.0" id="font2" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:bold;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="e" horiz-adv-x="55.615234" d="M37.2031 16.5 L50.875 14.2031 Q48.25 6.6875 42.5547 2.7578 Q36.8594 -1.1719 28.3281 -1.1719 Q14.7969 -1.1719 8.2969 7.6719 Q3.1719 14.75 3.1719 25.5312 Q3.1719 38.4219 9.9141 45.7266 Q16.6562 53.0312 26.9531 53.0312 Q38.5312 53.0312 45.2188 45.3906 Q51.9062 37.75 51.6094 21.9688 L17.2344 21.9688 Q17.3906 15.875 20.5625 12.4766 Q23.7344 9.0781 28.4688 9.0781 Q31.6875 9.0781 33.8828 10.8359 Q36.0781 12.5938 37.2031 16.5 ZM37.9844 30.375 Q37.8438 36.3281 34.9141 39.4297 Q31.9844 42.5312 27.7812 42.5312 Q23.2969 42.5312 20.3594 39.2656 Q17.4375 35.9844 17.4844 30.375 L37.9844 30.375 Z" + /><glyph unicode="g" horiz-adv-x="61.083984" d="M5.9062 -3.4219 L21.5781 -5.3281 Q21.9688 -8.0625 23.3906 -9.0781 Q25.3438 -10.5469 29.5469 -10.5469 Q34.9062 -10.5469 37.5938 -8.9375 Q39.4062 -7.8594 40.3281 -5.4688 Q40.9688 -3.7656 40.9688 0.8281 L40.9688 8.4062 Q34.8125 0 25.4375 0 Q14.9844 0 8.8906 8.8438 Q4.1094 15.8281 4.1094 26.2188 Q4.1094 39.2656 10.3828 46.1484 Q16.6562 53.0312 25.9844 53.0312 Q35.5938 53.0312 41.8438 44.5781 L41.8438 51.8594 L54.6875 51.8594 L54.6875 5.3281 Q54.6875 -3.8594 53.1719 -8.3984 Q51.6562 -12.9375 48.9219 -15.5234 Q46.1875 -18.1094 41.625 -19.5781 Q37.0625 -21.0469 30.0781 -21.0469 Q16.8906 -21.0469 11.375 -16.5312 Q5.8594 -12.0156 5.8594 -5.0781 Q5.8594 -4.3906 5.9062 -3.4219 ZM18.1719 27 Q18.1719 18.75 21.3672 14.9141 Q24.5625 11.0781 29.25 11.0781 Q34.2812 11.0781 37.75 15.0156 Q41.2188 18.9531 41.2188 26.6562 Q41.2188 34.7188 37.8984 38.625 Q34.5781 42.5312 29.5 42.5312 Q24.5625 42.5312 21.3672 38.6953 Q18.1719 34.8594 18.1719 27 Z" + /><glyph unicode="n" horiz-adv-x="61.083984" d="M54.3438 0 L40.625 0 L40.625 26.4688 Q40.625 34.8594 39.75 37.3281 Q38.875 39.7969 36.8906 41.1641 Q34.9062 42.5312 32.125 42.5312 Q28.5625 42.5312 25.7344 40.5781 Q22.9062 38.625 21.8516 35.3984 Q20.7969 32.1719 20.7969 23.4844 L20.7969 0 L7.0781 0 L7.0781 51.8594 L19.8281 51.8594 L19.8281 44.2344 Q26.6094 53.0312 36.9219 53.0312 Q41.4531 53.0312 45.2109 51.3906 Q48.9688 49.75 50.8984 47.2109 Q52.8281 44.6719 53.5859 41.4531 Q54.3438 38.2344 54.3438 32.2344 L54.3438 0 Z" + /><glyph unicode="a" horiz-adv-x="55.615234" d="M17.4375 36.0312 L4.9844 38.2812 Q7.0781 45.7969 12.2031 49.4141 Q17.3281 53.0312 27.4375 53.0312 Q36.625 53.0312 41.1172 50.8594 Q45.6094 48.6875 47.4375 45.3438 Q49.2656 42 49.2656 33.0625 L49.125 17.0469 Q49.125 10.2031 49.7812 6.9609 Q50.4375 3.7188 52.25 0 L38.6719 0 Q38.1406 1.375 37.3594 4.0469 Q37.0156 5.2812 36.8594 5.6719 Q33.3438 2.25 29.3438 0.5391 Q25.3438 -1.1719 20.7969 -1.1719 Q12.7969 -1.1719 8.1797 3.1719 Q3.5625 7.5156 3.5625 14.1562 Q3.5625 18.5625 5.6641 22 Q7.7656 25.4375 11.5469 27.2734 Q15.3281 29.1094 22.4688 30.4688 Q32.0781 32.2812 35.7969 33.8438 L35.7969 35.2031 Q35.7969 39.1562 33.8438 40.8438 Q31.8906 42.5312 26.4688 42.5312 Q22.7969 42.5312 20.75 41.0938 Q18.7031 39.6562 17.4375 36.0312 ZM35.7969 24.9062 Q33.1562 24.0312 27.4453 22.8047 Q21.7344 21.5781 19.9688 20.4062 Q17.2812 18.5 17.2812 15.5781 Q17.2812 12.7031 19.4297 10.6016 Q21.5781 8.5 24.9062 8.5 Q28.6094 8.5 31.9844 10.9375 Q34.4688 12.7969 35.25 15.4844 Q35.7969 17.2344 35.7969 22.1719 L35.7969 24.9062 Z" + /><glyph unicode="f" horiz-adv-x="33.30078" d="M1.1719 51.8594 L8.7969 51.8594 L8.7969 55.7656 Q8.7969 62.3125 10.1875 65.5312 Q11.5781 68.75 15.3125 70.7734 Q19.0469 72.7969 24.75 72.7969 Q30.6094 72.7969 36.2344 71.0469 L34.375 61.4688 Q31.1094 62.25 28.0781 62.25 Q25.0938 62.25 23.8047 60.8594 Q22.5156 59.4688 22.5156 55.5156 L22.5156 51.8594 L32.7656 51.8594 L32.7656 41.0625 L22.5156 41.0625 L22.5156 0 L8.7969 0 L8.7969 41.0625 L1.1719 41.0625 L1.1719 51.8594 Z" + /><glyph unicode="p" horiz-adv-x="61.083984" d="M6.7812 51.8594 L19.5781 51.8594 L19.5781 44.2344 Q22.0781 48.1406 26.3203 50.5859 Q30.5625 53.0312 35.75 53.0312 Q44.7812 53.0312 51.0781 45.9531 Q57.375 38.875 57.375 26.2188 Q57.375 13.2344 51.0234 6.0312 Q44.6719 -1.1719 35.6406 -1.1719 Q31.3438 -1.1719 27.8516 0.5391 Q24.3594 2.25 20.5156 6.3906 L20.5156 -19.7344 L6.7812 -19.7344 L6.7812 51.8594 ZM20.3594 26.8125 Q20.3594 18.0625 23.8281 13.8906 Q27.2969 9.7188 32.2812 9.7188 Q37.0625 9.7188 40.2344 13.5547 Q43.4062 17.3906 43.4062 26.125 Q43.4062 34.2812 40.1328 38.2344 Q36.8594 42.1875 32.0312 42.1875 Q27 42.1875 23.6797 38.3047 Q20.3594 34.4219 20.3594 26.8125 Z" + /><glyph unicode="m" horiz-adv-x="88.916016" d="M6.1562 51.8594 L18.7969 51.8594 L18.7969 44.7812 Q25.5938 53.0312 34.9688 53.0312 Q39.9375 53.0312 43.6016 50.9766 Q47.2656 48.9219 49.6094 44.7812 Q53.0312 48.9219 56.9844 50.9766 Q60.9375 53.0312 65.4375 53.0312 Q71.1406 53.0312 75.0938 50.7109 Q79.0469 48.3906 81 43.8906 Q82.4219 40.5781 82.4219 33.1562 L82.4219 0 L68.7031 0 L68.7031 29.6406 Q68.7031 37.3594 67.2812 39.5938 Q65.375 42.5312 61.4219 42.5312 Q58.5469 42.5312 56.0078 40.7734 Q53.4688 39.0156 52.3438 35.625 Q51.2188 32.2344 51.2188 24.9062 L51.2188 0 L37.5 0 L37.5 28.4219 Q37.5 35.9844 36.7656 38.1797 Q36.0312 40.375 34.4922 41.4531 Q32.9531 42.5312 30.3281 42.5312 Q27.1562 42.5312 24.6172 40.8203 Q22.0781 39.1094 20.9766 35.8906 Q19.875 32.6719 19.875 25.2031 L19.875 0 L6.1562 0 L6.1562 51.8594 Z" + /><glyph unicode="E" horiz-adv-x="66.69922" d="M7.2812 0 L7.2812 71.5781 L60.3594 71.5781 L60.3594 59.4688 L21.7344 59.4688 L21.7344 43.6094 L57.6719 43.6094 L57.6719 31.5469 L21.7344 31.5469 L21.7344 12.0625 L61.7188 12.0625 L61.7188 0 L7.2812 0 Z" + /></font + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >10</text + ></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >20</text + ></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >30</text + ></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >40</text + ></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >50</text + ></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >60</text + ></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >70</text + ></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >80</text + ></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >90</text + ></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="325" style="fill:none;" x1="73" x2="77.34" y1="325" + /><line y2="276" style="fill:none;" x1="73" x2="77.34" y1="276" + /><line y2="227" style="fill:none;" x1="73" x2="77.34" y1="227" + /><line y2="178" style="fill:none;" x1="73" x2="77.34" y1="178" + /><line y2="129" style="fill:none;" x1="73" x2="77.34" y1="129" + /><line y2="80" style="fill:none;" x1="73" x2="77.34" y1="80" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="325" style="fill:none;" x1="507" x2="502.66" y1="325" + /><line y2="276" style="fill:none;" x1="507" x2="502.66" y1="276" + /><line y2="227" style="fill:none;" x1="507" x2="502.66" y1="227" + /><line y2="178" style="fill:none;" x1="507" x2="502.66" y1="178" + /><line y2="129" style="fill:none;" x1="507" x2="502.66" y1="129" + /><line y2="80" style="fill:none;" x1="507" x2="502.66" y1="80" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >0</text + ></g + ><g transform="translate(67.6667,325)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-15" xml:space="preserve" y="5.5" style="stroke:none;" + >50</text + ></g + ><g transform="translate(67.6667,276)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >100</text + ></g + ><g transform="translate(67.6667,227)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >150</text + ></g + ><g transform="translate(67.6667,178)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >200</text + ></g + ><g transform="translate(67.6667,129)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >250</text + ></g + ><g transform="translate(67.6667,80)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >300</text + ></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-23" xml:space="preserve" y="5.5" style="stroke:none;" + >350</text + ></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><text x="-40" xml:space="preserve" y="-4" style="stroke:none;" + >Empfangen</text + ></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 75.1 L77.5684 261.7875 L82.1368 289.4718 L86.7053 363.9599 L91.2737 307.2254 L95.8421 362.0164 L100.4105 369.343 L104.979 354.3807 L109.5474 361.9688 L114.1158 362.5128 L118.6842 349.6222 L123.2526 358.0281 L127.8211 351.2132 L132.3895 357.7471 L136.9579 358.6982 L141.5263 341.2717 L146.0947 353.6311 L150.6632 355.0342 L155.2316 357.9427 L159.8 353.4687 L164.3684 355.3275 L168.9368 362.5093 L173.5053 343.5215 L178.0737 345.155 L182.6421 347.9055 L187.2105 356.3438 L191.7789 349.7505 L196.3474 366.6401 L200.9158 370.0116 L205.4842 347.3291 L210.0526 357.1107 L214.621 350.9171 L219.1895 352.1744 L223.7579 359.4998 L228.3263 348.958 L232.8947 341.2498 L237.4632 362.6831 L242.0316 353.3522 L246.6 345.781 L251.1684 361.8681 L255.7368 357.3319 L260.3053 354.6476 L264.8737 355.92 L269.4421 362.0248 L274.0105 366.566 L278.5789 368.0974 L283.1474 359.7258 L287.7158 370.2862 L292.2842 353.42 L296.8526 370.2862 L301.4211 359.7258 L305.9895 368.0974 L310.5579 366.566 L315.1263 362.0248 L319.6947 355.92 L324.2632 354.6476 L328.8316 357.3319 L333.4 361.8681 L337.9684 345.781 L342.5368 353.3522 L347.1053 362.6831 L351.6737 341.2498 L356.2421 348.958 L360.8105 359.4998 L365.3789 352.1744 L369.9474 350.9171 L374.5158 357.1107 L379.0842 347.3291 L383.6526 370.0116 L388.221 366.6401 L392.7895 349.7505 L397.3579 356.3438 L401.9263 347.9055 L406.4948 345.155 L411.0632 344.2163 L415.6316 362.5093 L420.2 357.2768 L424.7684 353.4687 L429.3369 357.9427 L433.9053 355.0342 L438.4737 353.6311 L443.0421 341.2717 L447.6105 358.6982 L452.179 357.7471 L456.7474 351.2132 L461.3158 358.0281 L465.8842 349.6222 L470.4526 362.5128 L475.0211 361.9688 L479.5895 354.3807 L484.1579 370.8774 L488.7263 362.0164 L493.2947 307.2254 L497.8632 363.9599 L502.4316 289.4718 L507 261.7875" style="fill:none; fill-rule:evenodd;" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/fig5.pdf b/buch/papers/reedsolomon/RS presentation/images/fig5.pdf Binary files differnew file mode 100644 index 0000000..37416a9 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig5.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig5.svg b/buch/papers/reedsolomon/RS presentation/images/fig5.svg new file mode 100644 index 0000000..7cfdb10 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig5.svg @@ -0,0 +1,121 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-2.6562 14 L-3.7969 14 L-3.7969 6.7188 Q-4.2188 7.1094 -4.8828 7.5078 Q-5.5469 7.9062 -6.0781 8.0938 L-6.0781 6.9844 Q-5.125 6.5469 -4.4062 5.9062 Q-3.6875 5.2656 -3.3906 4.6562 L-2.6562 4.6562 L-2.6562 14 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-0.9531 12.9062 L-0.9531 14 L-7.1094 14 Q-7.125 13.5938 -6.9688 13.2031 Q-6.7344 12.5781 -6.2188 11.9688 Q-5.7031 11.3594 -4.7188 10.5625 Q-3.2031 9.3125 -2.6719 8.5859 Q-2.1406 7.8594 -2.1406 7.2188 Q-2.1406 6.5312 -2.625 6.0703 Q-3.1094 5.6094 -3.8906 5.6094 Q-4.7188 5.6094 -5.2109 6.1016 Q-5.7031 6.5938 -5.7188 7.4688 L-6.8906 7.3438 Q-6.7656 6.0312 -5.9844 5.3438 Q-5.2031 4.6562 -3.875 4.6562 Q-2.5312 4.6562 -1.75 5.3984 Q-0.9688 6.1406 -0.9688 7.2344 Q-0.9688 7.7969 -1.1953 8.3359 Q-1.4219 8.875 -1.9531 9.4766 Q-2.4844 10.0781 -3.7188 11.1094 Q-4.75 11.9688 -5.0391 12.2812 Q-5.3281 12.5938 -5.5156 12.9062 L-0.9531 12.9062 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-6.9531 11.5469 L-5.8125 11.3906 Q-5.6094 12.3594 -5.1406 12.7891 Q-4.6719 13.2188 -3.9844 13.2188 Q-3.1875 13.2188 -2.6328 12.6641 Q-2.0781 12.1094 -2.0781 11.2812 Q-2.0781 10.4844 -2.5859 9.9766 Q-3.0938 9.4688 -3.8906 9.4688 Q-4.2188 9.4688 -4.7031 9.5938 L-4.5781 8.5938 Q-4.4531 8.6094 -4.3906 8.6094 Q-3.6562 8.6094 -3.0703 8.2266 Q-2.4844 7.8438 -2.4844 7.0469 Q-2.4844 6.4219 -2.9141 6.0078 Q-3.3438 5.5938 -4.0156 5.5938 Q-4.6875 5.5938 -5.125 6.0156 Q-5.5625 6.4375 -5.7031 7.2656 L-6.8438 7.0625 Q-6.625 5.9219 -5.8828 5.2891 Q-5.1406 4.6562 -4.0469 4.6562 Q-3.2812 4.6562 -2.6406 4.9844 Q-2 5.3125 -1.6562 5.875 Q-1.3125 6.4375 -1.3125 7.0781 Q-1.3125 7.6719 -1.6406 8.1719 Q-1.9688 8.6719 -2.5938 8.9531 Q-1.7812 9.1562 -1.3203 9.75 Q-0.8594 10.3438 -0.8594 11.25 Q-0.8594 12.4688 -1.75 13.3203 Q-2.6406 14.1719 -4 14.1719 Q-5.2188 14.1719 -6.0312 13.4375 Q-6.8438 12.7031 -6.9531 11.5469 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-3.2969 14 L-3.2969 11.7656 L-7.3281 11.7656 L-7.3281 10.7188 L-3.0938 4.6875 L-2.1562 4.6875 L-2.1562 10.7188 L-0.8906 10.7188 L-0.8906 11.7656 L-2.1562 11.7656 L-2.1562 14 L-3.2969 14 ZM-3.2969 10.7188 L-3.2969 6.5312 L-6.2188 10.7188 L-3.2969 10.7188 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-6.9531 11.5625 L-5.7656 11.4531 Q-5.625 12.3438 -5.1406 12.7812 Q-4.6562 13.2188 -3.9688 13.2188 Q-3.1406 13.2188 -2.5703 12.5938 Q-2 11.9688 -2 10.9531 Q-2 9.9688 -2.5469 9.4062 Q-3.0938 8.8438 -3.9844 8.8438 Q-4.5469 8.8438 -4.9922 9.0938 Q-5.4375 9.3438 -5.6875 9.7344 L-6.75 9.5938 L-5.8594 4.8281 L-1.2344 4.8281 L-1.2344 5.9062 L-4.9375 5.9062 L-5.4375 8.4062 Q-4.6094 7.8281 -3.6875 7.8281 Q-2.4688 7.8281 -1.6328 8.6719 Q-0.7969 9.5156 -0.7969 10.8438 Q-0.7969 12.1094 -1.5312 13.0312 Q-2.4219 14.1562 -3.9688 14.1562 Q-5.2344 14.1562 -6.0391 13.4453 Q-6.8438 12.7344 -6.9531 11.5625 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-1.0312 6.9688 L-2.1719 7.0625 Q-2.3125 6.3906 -2.5938 6.0781 Q-3.0625 5.5938 -3.7344 5.5938 Q-4.2812 5.5938 -4.7031 5.9062 Q-5.2344 6.2969 -5.5469 7.0469 Q-5.8594 7.7969 -5.875 9.2031 Q-5.4688 8.5781 -4.8672 8.2734 Q-4.2656 7.9688 -3.6094 7.9688 Q-2.4688 7.9688 -1.6641 8.8047 Q-0.8594 9.6406 -0.8594 10.9844 Q-0.8594 11.8594 -1.2422 12.6094 Q-1.625 13.3594 -2.2812 13.7578 Q-2.9375 14.1562 -3.7812 14.1562 Q-5.2031 14.1562 -6.1094 13.1094 Q-7.0156 12.0625 -7.0156 9.6406 Q-7.0156 6.9531 -6.0156 5.7188 Q-5.1406 4.6562 -3.6719 4.6562 Q-2.5781 4.6562 -1.875 5.2734 Q-1.1719 5.8906 -1.0312 6.9688 ZM-5.7031 10.9844 Q-5.7031 11.5781 -5.4531 12.1172 Q-5.2031 12.6562 -4.75 12.9375 Q-4.2969 13.2188 -3.7969 13.2188 Q-3.0781 13.2188 -2.5547 12.6328 Q-2.0312 12.0469 -2.0312 11.0469 Q-2.0312 10.0781 -2.5469 9.5234 Q-3.0625 8.9688 -3.8438 8.9688 Q-4.625 8.9688 -5.1641 9.5234 Q-5.7031 10.0781 -5.7031 10.9844 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-6.8906 5.9062 L-6.8906 4.8125 L-0.8594 4.8125 L-0.8594 5.7031 Q-1.75 6.6562 -2.625 8.2188 Q-3.5 9.7812 -3.9688 11.4375 Q-4.3125 12.6094 -4.4062 14 L-5.5781 14 Q-5.5625 12.9062 -5.1484 11.3516 Q-4.7344 9.7969 -3.9688 8.3516 Q-3.2031 6.9062 -2.3281 5.9062 L-6.8906 5.9062 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-5.2031 8.9531 Q-5.9062 8.6875 -6.25 8.2109 Q-6.5938 7.7344 -6.5938 7.0625 Q-6.5938 6.0469 -5.8672 5.3516 Q-5.1406 4.6562 -3.9219 4.6562 Q-2.7031 4.6562 -1.9609 5.3672 Q-1.2188 6.0781 -1.2188 7.0938 Q-1.2188 7.7344 -1.5625 8.2109 Q-1.9062 8.6875 -2.5938 8.9531 Q-1.7344 9.2344 -1.2891 9.8594 Q-0.8438 10.4844 -0.8438 11.3438 Q-0.8438 12.5312 -1.6875 13.3438 Q-2.5312 14.1562 -3.9062 14.1562 Q-5.2812 14.1562 -6.125 13.3438 Q-6.9688 12.5312 -6.9688 11.3125 Q-6.9688 10.4062 -6.5078 9.7891 Q-6.0469 9.1719 -5.2031 8.9531 ZM-5.4375 7.0156 Q-5.4375 7.6719 -5.0078 8.0938 Q-4.5781 8.5156 -3.9062 8.5156 Q-3.2344 8.5156 -2.8125 8.1016 Q-2.3906 7.6875 -2.3906 7.0781 Q-2.3906 6.4531 -2.8281 6.0234 Q-3.2656 5.5938 -3.9062 5.5938 Q-4.5625 5.5938 -5 6.0156 Q-5.4375 6.4375 -5.4375 7.0156 ZM-5.7969 11.3125 Q-5.7969 11.7969 -5.5625 12.2578 Q-5.3281 12.7188 -4.875 12.9688 Q-4.4219 13.2188 -3.8906 13.2188 Q-3.0781 13.2188 -2.5469 12.6953 Q-2.0156 12.1719 -2.0156 11.3594 Q-2.0156 10.5312 -2.5625 9.9922 Q-3.1094 9.4531 -3.9375 9.4531 Q-4.7344 9.4531 -5.2656 9.9844 Q-5.7969 10.5156 -5.7969 11.3125 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-6.7812 11.8438 L-5.6875 11.75 Q-5.5469 12.5156 -5.1562 12.8672 Q-4.7656 13.2188 -4.1406 13.2188 Q-3.625 13.2188 -3.2266 12.9766 Q-2.8281 12.7344 -2.5703 12.3359 Q-2.3125 11.9375 -2.1484 11.25 Q-1.9844 10.5625 -1.9844 9.8438 Q-1.9844 9.7656 -1.9844 9.625 Q-2.3281 10.1719 -2.9219 10.5078 Q-3.5156 10.8438 -4.2031 10.8438 Q-5.3594 10.8438 -6.1562 10.0078 Q-6.9531 9.1719 -6.9531 7.7969 Q-6.9531 6.375 -6.125 5.5156 Q-5.2969 4.6562 -4.0312 4.6562 Q-3.125 4.6562 -2.375 5.1484 Q-1.625 5.6406 -1.2344 6.5391 Q-0.8438 7.4375 -0.8438 9.1562 Q-0.8438 10.9375 -1.2266 11.9922 Q-1.6094 13.0469 -2.375 13.6016 Q-3.1406 14.1562 -4.1719 14.1562 Q-5.2656 14.1562 -5.9609 13.5547 Q-6.6562 12.9531 -6.7812 11.8438 ZM-2.1094 7.7344 Q-2.1094 6.75 -2.6328 6.1797 Q-3.1562 5.6094 -3.8906 5.6094 Q-4.6562 5.6094 -5.2188 6.2266 Q-5.7812 6.8438 -5.7812 7.8438 Q-5.7812 8.7188 -5.25 9.2734 Q-4.7188 9.8281 -3.9219 9.8281 Q-3.125 9.8281 -2.6172 9.2734 Q-2.1094 8.7188 -2.1094 7.7344 ZM0.4507 9.4062 Q0.4507 7.7656 0.7866 6.7578 Q1.1225 5.75 1.7944 5.2031 Q2.4663 4.6562 3.4819 4.6562 Q4.2319 4.6562 4.7944 4.9609 Q5.3569 5.2656 5.7241 5.8281 Q6.0913 6.3906 6.3022 7.2109 Q6.5132 8.0312 6.5132 9.4062 Q6.5132 11.0469 6.1772 12.0547 Q5.8413 13.0625 5.1694 13.6094 Q4.4975 14.1562 3.4819 14.1562 Q2.1382 14.1562 1.3569 13.1875 Q0.4507 12.0312 0.4507 9.4062 ZM1.6225 9.4062 Q1.6225 11.7031 2.1538 12.4609 Q2.685 13.2188 3.4819 13.2188 Q4.2632 13.2188 4.8022 12.4609 Q5.3413 11.7031 5.3413 9.4062 Q5.3413 7.1094 4.8022 6.3594 Q4.2632 5.6094 3.4663 5.6094 Q2.685 5.6094 2.2007 6.2656 Q1.6225 7.125 1.6225 9.4062 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="316.8333" style="fill:none;" x1="73" x2="77.34" y1="316.8333" + /><line y2="259.6667" style="fill:none;" x1="73" x2="77.34" y1="259.6667" + /><line y2="202.5" style="fill:none;" x1="73" x2="77.34" y1="202.5" + /><line y2="145.3333" style="fill:none;" x1="73" x2="77.34" y1="145.3333" + /><line y2="88.1667" style="fill:none;" x1="73" x2="77.34" y1="88.1667" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="316.8333" style="fill:none;" x1="507" x2="502.66" y1="316.8333" + /><line y2="259.6667" style="fill:none;" x1="507" x2="502.66" y1="259.6667" + /><line y2="202.5" style="fill:none;" x1="507" x2="502.66" y1="202.5" + /><line y2="145.3333" style="fill:none;" x1="507" x2="502.66" y1="145.3333" + /><line y2="88.1667" style="fill:none;" x1="507" x2="502.66" y1="88.1667" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-7.4531 0.9062 Q-7.4531 -0.7344 -7.1172 -1.7422 Q-6.7812 -2.75 -6.1094 -3.2969 Q-5.4375 -3.8438 -4.4219 -3.8438 Q-3.6719 -3.8438 -3.1094 -3.5391 Q-2.5469 -3.2344 -2.1797 -2.6719 Q-1.8125 -2.1094 -1.6016 -1.2891 Q-1.3906 -0.4688 -1.3906 0.9062 Q-1.3906 2.5469 -1.7266 3.5547 Q-2.0625 4.5625 -2.7344 5.1094 Q-3.4062 5.6562 -4.4219 5.6562 Q-5.7656 5.6562 -6.5469 4.6875 Q-7.4531 3.5312 -7.4531 0.9062 ZM-6.2812 0.9062 Q-6.2812 3.2031 -5.75 3.9609 Q-5.2188 4.7188 -4.4219 4.7188 Q-3.6406 4.7188 -3.1016 3.9609 Q-2.5625 3.2031 -2.5625 0.9062 Q-2.5625 -1.3906 -3.1016 -2.1406 Q-3.6406 -2.8906 -4.4375 -2.8906 Q-5.2188 -2.8906 -5.7031 -2.2344 Q-6.2812 -1.375 -6.2812 0.9062 Z" + /></g + ><g transform="translate(67.6667,316.8333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-1.4531 4.4062 L-1.4531 5.5 L-7.6094 5.5 Q-7.625 5.0938 -7.4688 4.7031 Q-7.2344 4.0781 -6.7188 3.4688 Q-6.2031 2.8594 -5.2188 2.0625 Q-3.7031 0.8125 -3.1719 0.0859 Q-2.6406 -0.6406 -2.6406 -1.2812 Q-2.6406 -1.9688 -3.125 -2.4297 Q-3.6094 -2.8906 -4.3906 -2.8906 Q-5.2188 -2.8906 -5.7109 -2.3984 Q-6.2031 -1.9062 -6.2188 -1.0312 L-7.3906 -1.1562 Q-7.2656 -2.4688 -6.4844 -3.1562 Q-5.7031 -3.8438 -4.375 -3.8438 Q-3.0312 -3.8438 -2.25 -3.1016 Q-1.4688 -2.3594 -1.4688 -1.2656 Q-1.4688 -0.7031 -1.6953 -0.1641 Q-1.9219 0.375 -2.4531 0.9766 Q-2.9844 1.5781 -4.2188 2.6094 Q-5.25 3.4688 -5.5391 3.7812 Q-5.8281 4.0938 -6.0156 4.4062 L-1.4531 4.4062 Z" + /></g + ><g transform="translate(67.6667,259.6667)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-3.7969 5.5 L-3.7969 3.2656 L-7.8281 3.2656 L-7.8281 2.2188 L-3.5938 -3.8125 L-2.6562 -3.8125 L-2.6562 2.2188 L-1.3906 2.2188 L-1.3906 3.2656 L-2.6562 3.2656 L-2.6562 5.5 L-3.7969 5.5 ZM-3.7969 2.2188 L-3.7969 -1.9688 L-6.7188 2.2188 L-3.7969 2.2188 Z" + /></g + ><g transform="translate(67.6667,202.5)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-1.5312 -1.5312 L-2.6719 -1.4375 Q-2.8125 -2.1094 -3.0938 -2.4219 Q-3.5625 -2.9062 -4.2344 -2.9062 Q-4.7812 -2.9062 -5.2031 -2.5938 Q-5.7344 -2.2031 -6.0469 -1.4531 Q-6.3594 -0.7031 -6.375 0.7031 Q-5.9688 0.0781 -5.3672 -0.2266 Q-4.7656 -0.5312 -4.1094 -0.5312 Q-2.9688 -0.5312 -2.1641 0.3047 Q-1.3594 1.1406 -1.3594 2.4844 Q-1.3594 3.3594 -1.7422 4.1094 Q-2.125 4.8594 -2.7812 5.2578 Q-3.4375 5.6562 -4.2812 5.6562 Q-5.7031 5.6562 -6.6094 4.6094 Q-7.5156 3.5625 -7.5156 1.1406 Q-7.5156 -1.5469 -6.5156 -2.7812 Q-5.6406 -3.8438 -4.1719 -3.8438 Q-3.0781 -3.8438 -2.375 -3.2266 Q-1.6719 -2.6094 -1.5312 -1.5312 ZM-6.2031 2.4844 Q-6.2031 3.0781 -5.9531 3.6172 Q-5.7031 4.1562 -5.25 4.4375 Q-4.7969 4.7188 -4.2969 4.7188 Q-3.5781 4.7188 -3.0547 4.1328 Q-2.5312 3.5469 -2.5312 2.5469 Q-2.5312 1.5781 -3.0469 1.0234 Q-3.5625 0.4688 -4.3438 0.4688 Q-5.125 0.4688 -5.6641 1.0234 Q-6.2031 1.5781 -6.2031 2.4844 Z" + /></g + ><g transform="translate(67.6667,145.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-5.7031 0.4531 Q-6.4062 0.1875 -6.75 -0.2891 Q-7.0938 -0.7656 -7.0938 -1.4375 Q-7.0938 -2.4531 -6.3672 -3.1484 Q-5.6406 -3.8438 -4.4219 -3.8438 Q-3.2031 -3.8438 -2.4609 -3.1328 Q-1.7188 -2.4219 -1.7188 -1.4062 Q-1.7188 -0.7656 -2.0625 -0.2891 Q-2.4062 0.1875 -3.0938 0.4531 Q-2.2344 0.7344 -1.7891 1.3594 Q-1.3438 1.9844 -1.3438 2.8438 Q-1.3438 4.0312 -2.1875 4.8438 Q-3.0312 5.6562 -4.4062 5.6562 Q-5.7812 5.6562 -6.625 4.8438 Q-7.4688 4.0312 -7.4688 2.8125 Q-7.4688 1.9062 -7.0078 1.2891 Q-6.5469 0.6719 -5.7031 0.4531 ZM-5.9375 -1.4844 Q-5.9375 -0.8281 -5.5078 -0.4062 Q-5.0781 0.0156 -4.4062 0.0156 Q-3.7344 0.0156 -3.3125 -0.3984 Q-2.8906 -0.8125 -2.8906 -1.4219 Q-2.8906 -2.0469 -3.3281 -2.4766 Q-3.7656 -2.9062 -4.4062 -2.9062 Q-5.0625 -2.9062 -5.5 -2.4844 Q-5.9375 -2.0625 -5.9375 -1.4844 ZM-6.2969 2.8125 Q-6.2969 3.2969 -6.0625 3.7578 Q-5.8281 4.2188 -5.375 4.4688 Q-4.9219 4.7188 -4.3906 4.7188 Q-3.5781 4.7188 -3.0469 4.1953 Q-2.5156 3.6719 -2.5156 2.8594 Q-2.5156 2.0312 -3.0625 1.4922 Q-3.6094 0.9531 -4.4375 0.9531 Q-5.2344 0.9531 -5.7656 1.4844 Q-6.2969 2.0156 -6.2969 2.8125 Z" + /></g + ><g transform="translate(67.6667,88.1667)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-10.1562 5.5 L-11.2969 5.5 L-11.2969 -1.7812 Q-11.7188 -1.3906 -12.3828 -0.9922 Q-13.0469 -0.5938 -13.5781 -0.4062 L-13.5781 -1.5156 Q-12.625 -1.9531 -11.9062 -2.5938 Q-11.1875 -3.2344 -10.8906 -3.8438 L-10.1562 -3.8438 L-10.1562 5.5 ZM-7.0493 0.9062 Q-7.0493 -0.7344 -6.7134 -1.7422 Q-6.3775 -2.75 -5.7056 -3.2969 Q-5.0337 -3.8438 -4.0181 -3.8438 Q-3.2681 -3.8438 -2.7056 -3.5391 Q-2.1431 -3.2344 -1.7759 -2.6719 Q-1.4087 -2.1094 -1.1978 -1.2891 Q-0.9868 -0.4688 -0.9868 0.9062 Q-0.9868 2.5469 -1.3228 3.5547 Q-1.6587 4.5625 -2.3306 5.1094 Q-3.0025 5.6562 -4.0181 5.6562 Q-5.3618 5.6562 -6.1431 4.6875 Q-7.0493 3.5312 -7.0493 0.9062 ZM-5.8775 0.9062 Q-5.8775 3.2031 -5.3462 3.9609 Q-4.815 4.7188 -4.0181 4.7188 Q-3.2368 4.7188 -2.6978 3.9609 Q-2.1587 3.2031 -2.1587 0.9062 Q-2.1587 -1.3906 -2.6978 -2.1406 Q-3.2368 -2.8906 -4.0337 -2.8906 Q-4.815 -2.8906 -5.2993 -2.2344 Q-5.8775 -1.375 -5.8775 0.9062 Z" + /></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><path style="stroke:none;" d="M-10.1562 5.5 L-11.2969 5.5 L-11.2969 -1.7812 Q-11.7188 -1.3906 -12.3828 -0.9922 Q-13.0469 -0.5938 -13.5781 -0.4062 L-13.5781 -1.5156 Q-12.625 -1.9531 -11.9062 -2.5938 Q-11.1875 -3.2344 -10.8906 -3.8438 L-10.1562 -3.8438 L-10.1562 5.5 ZM-1.0493 4.4062 L-1.0493 5.5 L-7.2056 5.5 Q-7.2212 5.0938 -7.065 4.7031 Q-6.8306 4.0781 -6.315 3.4688 Q-5.7993 2.8594 -4.815 2.0625 Q-3.2993 0.8125 -2.7681 0.0859 Q-2.2368 -0.6406 -2.2368 -1.2812 Q-2.2368 -1.9688 -2.7212 -2.4297 Q-3.2056 -2.8906 -3.9868 -2.8906 Q-4.815 -2.8906 -5.3072 -2.3984 Q-5.7993 -1.9062 -5.815 -1.0312 L-6.9868 -1.1562 Q-6.8618 -2.4688 -6.0806 -3.1562 Q-5.2993 -3.8438 -3.9712 -3.8438 Q-2.6275 -3.8438 -1.8462 -3.1016 Q-1.065 -2.3594 -1.065 -1.2656 Q-1.065 -0.7031 -1.2915 -0.1641 Q-1.5181 0.375 -2.0493 0.9766 Q-2.5806 1.5781 -3.815 2.6094 Q-4.8462 3.4688 -5.1353 3.7812 Q-5.4243 4.0938 -5.6118 4.4062 L-1.0493 4.4062 Z" + /></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><path style="stroke:none;" d="M-32.9219 -14.7344 L-28.9531 -14.7344 Q-27.6094 -14.7344 -26.9062 -14.5312 Q-25.9688 -14.25 -25.2969 -13.5391 Q-24.625 -12.8281 -24.2734 -11.8047 Q-23.9219 -10.7812 -23.9219 -9.2656 Q-23.9219 -7.9375 -24.25 -6.9844 Q-24.6406 -5.8125 -25.3906 -5.0781 Q-25.9531 -4.5312 -26.9219 -4.2344 Q-27.6406 -4 -28.8438 -4 L-32.9219 -4 L-32.9219 -14.7344 ZM-30.75 -12.9219 L-30.75 -5.8125 L-29.125 -5.8125 Q-28.2188 -5.8125 -27.8125 -5.9062 Q-27.2969 -6.0469 -26.9453 -6.3594 Q-26.5938 -6.6719 -26.375 -7.3906 Q-26.1562 -8.1094 -26.1562 -9.3594 Q-26.1562 -10.6094 -26.375 -11.2734 Q-26.5938 -11.9375 -26.9922 -12.3125 Q-27.3906 -12.6875 -28 -12.8125 Q-28.4531 -12.9219 -29.7812 -12.9219 L-30.75 -12.9219 ZM-17.8376 -6.4688 L-15.7907 -6.125 Q-16.1814 -5 -17.0329 -4.4141 Q-17.8845 -3.8281 -19.1657 -3.8281 Q-21.197 -3.8281 -22.1657 -5.1562 Q-22.947 -6.2188 -22.947 -7.8281 Q-22.947 -9.7656 -21.9314 -10.8594 Q-20.9157 -11.9531 -19.3689 -11.9531 Q-17.6345 -11.9531 -16.6345 -10.8047 Q-15.6345 -9.6562 -15.6814 -7.2969 L-20.8376 -7.2969 Q-20.8064 -6.375 -20.3298 -5.8672 Q-19.8532 -5.3594 -19.1501 -5.3594 Q-18.6657 -5.3594 -18.3376 -5.625 Q-18.0095 -5.8906 -17.8376 -6.4688 ZM-17.7126 -8.5625 Q-17.7439 -9.4531 -18.1814 -9.9141 Q-18.6189 -10.375 -19.2439 -10.375 Q-19.9157 -10.375 -20.3689 -9.8906 Q-20.8064 -9.3906 -20.7907 -8.5625 L-17.7126 -8.5625 ZM-7.4052 -9.4844 L-9.4365 -9.1094 Q-9.5302 -9.7188 -9.8974 -10.0234 Q-10.2646 -10.3281 -10.8427 -10.3281 Q-11.6084 -10.3281 -12.0693 -9.8047 Q-12.5302 -9.2812 -12.5302 -8.0312 Q-12.5302 -6.6406 -12.0615 -6.0703 Q-11.5927 -5.5 -10.8115 -5.5 Q-10.2177 -5.5 -9.8506 -5.8359 Q-9.4834 -6.1719 -9.3271 -6.9844 L-7.2959 -6.6406 Q-7.624 -5.25 -8.5146 -4.5391 Q-9.4052 -3.8281 -10.9052 -3.8281 Q-12.6084 -3.8281 -13.624 -4.9062 Q-14.6396 -5.9844 -14.6396 -7.875 Q-14.6396 -9.8125 -13.624 -10.8828 Q-12.6084 -11.9531 -10.874 -11.9531 Q-9.4521 -11.9531 -8.6084 -11.3438 Q-7.7646 -10.7344 -7.4052 -9.4844 ZM-6.5198 -8 Q-6.5198 -9.0312 -6.0119 -9.9844 Q-5.5041 -10.9375 -4.5744 -11.4453 Q-3.6448 -11.9531 -2.5041 -11.9531 Q-0.7385 -11.9531 0.3865 -10.8047 Q1.5115 -9.6562 1.5115 -7.9062 Q1.5115 -6.1406 0.3709 -4.9844 Q-0.7698 -3.8281 -2.4885 -3.8281 Q-3.5666 -3.8281 -4.5354 -4.3125 Q-5.5041 -4.7969 -6.0119 -5.7266 Q-6.5198 -6.6562 -6.5198 -8 ZM-4.4104 -7.8906 Q-4.4104 -6.7344 -3.8557 -6.1172 Q-3.301 -5.5 -2.5041 -5.5 Q-1.6916 -5.5 -1.1448 -6.1172 Q-0.5979 -6.7344 -0.5979 -7.9062 Q-0.5979 -9.0469 -1.1448 -9.6641 Q-1.6916 -10.2812 -2.5041 -10.2812 Q-3.301 -10.2812 -3.8557 -9.6641 Q-4.4104 -9.0469 -4.4104 -7.8906 ZM10.0422 -4 L8.136 -4 L8.136 -5.1406 Q7.6672 -4.4688 7.0188 -4.1484 Q6.3704 -3.8281 5.7141 -3.8281 Q4.3704 -3.8281 3.4094 -4.9062 Q2.4485 -5.9844 2.4485 -7.9219 Q2.4485 -9.8906 3.3782 -10.9219 Q4.3079 -11.9531 5.7297 -11.9531 Q7.0422 -11.9531 7.9954 -10.875 L7.9954 -14.7344 L10.0422 -14.7344 L10.0422 -4 ZM4.5579 -8.0625 Q4.5579 -6.8125 4.9016 -6.25 Q5.4016 -5.4531 6.2922 -5.4531 Q6.9954 -5.4531 7.4954 -6.0547 Q7.9954 -6.6562 7.9954 -7.8594 Q7.9954 -9.2031 7.511 -9.7891 Q7.0266 -10.375 6.2766 -10.375 Q5.5422 -10.375 5.05 -9.7969 Q4.5579 -9.2188 4.5579 -8.0625 ZM11.8699 -12.8281 L11.8699 -14.7344 L13.9324 -14.7344 L13.9324 -12.8281 L11.8699 -12.8281 ZM11.8699 -4 L11.8699 -11.7812 L13.9324 -11.7812 L13.9324 -4 L11.8699 -4 ZM20.4418 -6.4688 L22.4887 -6.125 Q22.0981 -5 21.2465 -4.4141 Q20.395 -3.8281 19.1137 -3.8281 Q17.0825 -3.8281 16.1137 -5.1562 Q15.3325 -6.2188 15.3325 -7.8281 Q15.3325 -9.7656 16.3481 -10.8594 Q17.3637 -11.9531 18.9106 -11.9531 Q20.645 -11.9531 21.645 -10.8047 Q22.645 -9.6562 22.5981 -7.2969 L17.4418 -7.2969 Q17.4731 -6.375 17.9496 -5.8672 Q18.4262 -5.3594 19.1293 -5.3594 Q19.6137 -5.3594 19.9418 -5.625 Q20.27 -5.8906 20.4418 -6.4688 ZM20.5668 -8.5625 Q20.5356 -9.4531 20.0981 -9.9141 Q19.6606 -10.375 19.0356 -10.375 Q18.3637 -10.375 17.9106 -9.8906 Q17.4731 -9.3906 17.4887 -8.5625 L20.5668 -8.5625 ZM26.0617 -4 L23.9992 -4 L23.9992 -11.7812 L25.9211 -11.7812 L25.9211 -10.6719 Q26.4054 -11.4531 26.7961 -11.7031 Q27.1867 -11.9531 27.6867 -11.9531 Q28.3898 -11.9531 29.0461 -11.5625 L28.4054 -9.7656 Q27.8898 -10.1094 27.4367 -10.1094 Q26.9992 -10.1094 26.7023 -9.8672 Q26.4054 -9.625 26.2336 -9.0078 Q26.0617 -8.3906 26.0617 -6.4062 L26.0617 -4 ZM33.3591 -11.7812 L33.3591 -10.1406 L31.9528 -10.1406 L31.9528 -7 Q31.9528 -6.0469 31.9919 -5.8906 Q32.0309 -5.7344 32.1794 -5.6328 Q32.3278 -5.5312 32.5309 -5.5312 Q32.8122 -5.5312 33.3591 -5.7344 L33.5309 -4.125 Q32.8122 -3.8281 31.9059 -3.8281 Q31.3434 -3.8281 30.8981 -4.0156 Q30.4528 -4.2031 30.2419 -4.5 Q30.0309 -4.7969 29.9528 -5.2969 Q29.8903 -5.6562 29.8903 -6.75 L29.8903 -10.1406 L28.9528 -10.1406 L28.9528 -11.7812 L29.8903 -11.7812 L29.8903 -13.3281 L31.9528 -14.5312 L31.9528 -11.7812 L33.3591 -11.7812 Z" + /></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 229.5946 L77.5684 258.9224 L82.1368 260.0471 L86.7053 288.5562 L91.2737 230.5276 L95.8421 230.554 L100.4105 145.9635 L104.979 146.6942 L109.5474 203.2442 L114.1158 259.5181 L118.6842 145.3154 L123.2526 231.7605 L127.8211 260.2617 L132.3895 116.3788 L136.9579 115.7342 L141.5263 201.8038 L146.0947 202.3494 L150.6632 202.2886 L155.2316 316.2004 L159.8 287.628 L164.3684 231.0379 L168.9368 317.3546 L173.5053 145.9825 L178.0737 317.2646 L182.6421 317.1311 L187.2105 317.2646 L191.7789 317.4825 L196.3474 260.1949 L200.9158 288.2009 L205.4842 116.1281 L210.0526 259.0344 L214.621 316.6216 L219.1895 116.6 L223.7579 87.4705 L228.3263 258.6484 L232.8947 345.0273 L237.4632 288.845 L242.0316 260.3438 L246.6 202.4819 L251.1684 288.1006 L255.7368 203.2442 L260.3053 175.2774 L264.8737 317.4592 L269.4421 344.8843 L274.0105 287.6941 L278.5789 288.5562 L283.1474 260.0471 L287.7158 230.3393 L292.2842 343.9279 L296.8526 287.5054 L301.4211 145.7147 L305.9895 373.4924 L310.5579 116.1943 L315.1263 173.3876 L319.6947 231.7126 L324.2632 203.8606 L328.8316 317.5771 L333.4 230.9353 L337.9684 88.1488 L342.5368 231.7605 L347.1053 231.6785 L351.6737 316.4541 L356.2421 230.066 L360.8105 201.8038 L365.3789 373.2122 L369.9474 373.6689 L374.5158 373.2603 L379.0842 373.358 L383.6526 372.7393 L388.221 372.6437 L392.7895 373.3492 L397.3579 373.2242 L401.9263 373.7023 L406.4948 373.2242 L411.0632 373.3492 L415.6316 372.6437 L420.2 372.7393 L424.7684 373.358 L429.3369 373.2603 L433.9053 373.6689 L438.4737 373.2122 L443.0421 373.2993 L447.6105 372.5714 L452.179 372.8526 L456.7474 373.3342 L461.3158 373.3198 L465.8842 373.5887 L470.4526 373.2233 L475.0211 373.2122 L479.5895 372.5265 L484.1579 372.976 L488.7263 373.294 L493.2947 373.3992 L497.8632 373.4924 L502.4316 373.2531 L507 373.0999" style="fill:none; fill-rule:evenodd;" + /></g + ><g transform="translate(0,420)" style="fill:gray; text-rendering:geometricPrecision; color-rendering:optimizeQuality; image-rendering:optimizeQuality; color-interpolation:linearRGB; stroke:gray;" + ><circle r="2.6667" style="stroke:none;" cx="0" cy="0" + /></g + ><g transform="translate(0,420)" style="stroke-linecap:butt; fill:rgb(255,255,220); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(255,255,220); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path style="stroke:none;" d="M0 -2.6667 C1.4728 -2.6667 2.6667 -1.4728 2.6667 0 L2.6667 0 C2.6667 1.4728 1.4728 2.6667 -0 2.6667 C-1.4728 2.6667 -2.6667 1.4728 -2.6667 -0 C-2.6667 -1.4728 -1.4728 -2.6667 0 -2.6667 ZM0 -4 C-2.2091 -4 -4 -2.2091 -4 -0 C-4 2.2091 -2.2091 4 -0 4 C2.2091 4 4 2.2091 4 0 L4 0 C4 -2.2091 2.2091 -4 0 -4 Z" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/fig6.pdf b/buch/papers/reedsolomon/RS presentation/images/fig6.pdf Binary files differnew file mode 100644 index 0000000..61cbc01 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig6.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig6.svg b/buch/papers/reedsolomon/RS presentation/images/fig6.svg new file mode 100644 index 0000000..f8f8369 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig6.svg @@ -0,0 +1,158 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ><font horiz-adv-x="75.0" id="font1" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:normal;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="0" horiz-adv-x="55.615234" d="M4.1562 35.2969 Q4.1562 48 6.7656 55.7422 Q9.375 63.4844 14.5234 67.6797 Q19.6719 71.875 27.4844 71.875 Q33.25 71.875 37.5938 69.5547 Q41.9375 67.2344 44.7734 62.8672 Q47.6094 58.5 49.2188 52.2266 Q50.8281 45.9531 50.8281 35.2969 Q50.8281 22.7031 48.2422 14.9688 Q45.6562 7.2344 40.5078 3.0078 Q35.3594 -1.2188 27.4844 -1.2188 Q17.1406 -1.2188 11.2344 6.2031 Q4.1562 15.1406 4.1562 35.2969 ZM13.1875 35.2969 Q13.1875 17.6719 17.3125 11.8359 Q21.4375 6 27.4844 6 Q33.5469 6 37.6719 11.8594 Q41.7969 17.7188 41.7969 35.2969 Q41.7969 52.9844 37.6719 58.7891 Q33.5469 64.5938 27.3906 64.5938 Q21.3438 64.5938 17.7188 59.4688 Q13.1875 52.9375 13.1875 35.2969 Z" + /><glyph unicode="1" horiz-adv-x="55.615234" d="M37.25 0 L28.4688 0 L28.4688 56 Q25.2969 52.9844 20.1406 49.9531 Q14.9844 46.9219 10.8906 45.4062 L10.8906 53.9062 Q18.2656 57.375 23.7812 62.3047 Q29.2969 67.2344 31.5938 71.875 L37.25 71.875 L37.25 0 Z" + /><glyph unicode="2" horiz-adv-x="55.615234" d="M50.3438 8.4531 L50.3438 0 L3.0312 0 Q2.9375 3.1719 4.0469 6.1094 Q5.8594 10.9375 9.8359 15.625 Q13.8125 20.3125 21.3438 26.4688 Q33.0156 36.0312 37.1172 41.625 Q41.2188 47.2188 41.2188 52.2031 Q41.2188 57.4219 37.4766 61.0078 Q33.7344 64.5938 27.7344 64.5938 Q21.3906 64.5938 17.5781 60.7891 Q13.7656 56.9844 13.7188 50.25 L4.6875 51.1719 Q5.6094 61.2812 11.6641 66.5781 Q17.7188 71.875 27.9375 71.875 Q38.2344 71.875 44.2422 66.1641 Q50.25 60.4531 50.25 52 Q50.25 47.7031 48.4922 43.5547 Q46.7344 39.4062 42.6562 34.8125 Q38.5781 30.2188 29.1094 22.2188 Q21.1875 15.5781 18.9453 13.2109 Q16.7031 10.8438 15.2344 8.4531 L50.3438 8.4531 Z" + /><glyph unicode="3" horiz-adv-x="55.615234" d="M4.2031 18.8906 L12.9844 20.0625 Q14.5 12.5938 18.1406 9.2969 Q21.7812 6 27 6 Q33.2031 6 37.4766 10.2969 Q41.75 14.5938 41.75 20.9531 Q41.75 27 37.7969 30.9297 Q33.8438 34.8594 27.7344 34.8594 Q25.25 34.8594 21.5312 33.8906 L22.5156 41.6094 Q23.3906 41.5 23.9219 41.5 Q29.5469 41.5 34.0391 44.4297 Q38.5312 47.3594 38.5312 53.4688 Q38.5312 58.2969 35.2578 61.4766 Q31.9844 64.6562 26.8125 64.6562 Q21.6875 64.6562 18.2656 61.4297 Q14.8438 58.2031 13.875 51.7656 L5.0781 53.3281 Q6.6875 62.1562 12.3984 67.0156 Q18.1094 71.875 26.6094 71.875 Q32.4688 71.875 37.3984 69.3594 Q42.3281 66.8438 44.9453 62.5 Q47.5625 58.1562 47.5625 53.2656 Q47.5625 48.6406 45.0703 44.8281 Q42.5781 41.0156 37.7031 38.7656 Q44.0469 37.3125 47.5625 32.6953 Q51.0781 28.0781 51.0781 21.1406 Q51.0781 11.7656 44.2422 5.25 Q37.4062 -1.2656 26.9531 -1.2656 Q17.5312 -1.2656 11.3047 4.3516 Q5.0781 9.9688 4.2031 18.8906 Z" + /><glyph unicode="4" horiz-adv-x="55.615234" d="M32.3281 0 L32.3281 17.1406 L1.2656 17.1406 L1.2656 25.2031 L33.9375 71.5781 L41.1094 71.5781 L41.1094 25.2031 L50.7812 25.2031 L50.7812 17.1406 L41.1094 17.1406 L41.1094 0 L32.3281 0 ZM32.3281 25.2031 L32.3281 57.4688 L9.9062 25.2031 L32.3281 25.2031 Z" + /><glyph unicode="5" horiz-adv-x="55.615234" d="M4.1562 18.75 L13.375 19.5312 Q14.4062 12.7969 18.1406 9.3984 Q21.875 6 27.1562 6 Q33.5 6 37.8906 10.7891 Q42.2812 15.5781 42.2812 23.4844 Q42.2812 31 38.0625 35.3516 Q33.8438 39.7031 27 39.7031 Q22.75 39.7031 19.3359 37.7734 Q15.9219 35.8438 13.9688 32.7656 L5.7188 33.8438 L12.6406 70.6094 L48.25 70.6094 L48.25 62.2031 L19.6719 62.2031 L15.8281 42.9688 Q22.2656 47.4688 29.3438 47.4688 Q38.7188 47.4688 45.1641 40.9688 Q51.6094 34.4688 51.6094 24.2656 Q51.6094 14.5469 45.9531 7.4688 Q39.0625 -1.2188 27.1562 -1.2188 Q17.3906 -1.2188 11.2109 4.25 Q5.0312 9.7188 4.1562 18.75 Z" + /><glyph unicode="6" horiz-adv-x="55.615234" d="M49.75 54.0469 L41.0156 53.375 Q39.8438 58.5469 37.7031 60.8906 Q34.125 64.6562 28.9062 64.6562 Q24.7031 64.6562 21.5312 62.3125 Q17.3906 59.2812 14.9922 53.4688 Q12.5938 47.6562 12.5 36.9219 Q15.6719 41.75 20.2656 44.0938 Q24.8594 46.4375 29.8906 46.4375 Q38.6719 46.4375 44.8516 39.9688 Q51.0312 33.5 51.0312 23.25 Q51.0312 16.5 48.125 10.7188 Q45.2188 4.9375 40.1406 1.8594 Q35.0625 -1.2188 28.6094 -1.2188 Q17.625 -1.2188 10.6953 6.8594 Q3.7656 14.9375 3.7656 33.5 Q3.7656 54.25 11.4219 63.6719 Q18.1094 71.875 29.4375 71.875 Q37.8906 71.875 43.2891 67.1406 Q48.6875 62.4062 49.75 54.0469 ZM13.875 23.1875 Q13.875 18.6562 15.7969 14.5078 Q17.7188 10.3594 21.1875 8.1797 Q24.6562 6 28.4688 6 Q34.0312 6 38.0391 10.4922 Q42.0469 14.9844 42.0469 22.7031 Q42.0469 30.125 38.0859 34.3984 Q34.125 38.6719 28.125 38.6719 Q22.1719 38.6719 18.0234 34.3984 Q13.875 30.125 13.875 23.1875 Z" + /><glyph unicode="7" horiz-adv-x="55.615234" d="M4.7344 62.2031 L4.7344 70.6562 L51.0781 70.6562 L51.0781 63.8125 Q44.2344 56.5469 37.5234 44.4844 Q30.8125 32.4219 27.1562 19.6719 Q24.5156 10.6875 23.7812 0 L14.75 0 Q14.8906 8.4531 18.0625 20.4141 Q21.2344 32.375 27.1719 43.4844 Q33.1094 54.5938 39.7969 62.2031 L4.7344 62.2031 Z" + /><glyph unicode="8" horiz-adv-x="55.615234" d="M17.6719 38.8125 Q12.2031 40.8281 9.5703 44.5391 Q6.9375 48.25 6.9375 53.4219 Q6.9375 61.2344 12.5547 66.5547 Q18.1719 71.875 27.4844 71.875 Q36.8594 71.875 42.5781 66.4297 Q48.2969 60.9844 48.2969 53.1719 Q48.2969 48.1875 45.6797 44.5078 Q43.0625 40.8281 37.75 38.8125 Q44.3438 36.6719 47.7812 31.8828 Q51.2188 27.0938 51.2188 20.4531 Q51.2188 11.2812 44.7266 5.0312 Q38.2344 -1.2188 27.6406 -1.2188 Q17.0469 -1.2188 10.5469 5.0547 Q4.0469 11.3281 4.0469 20.7031 Q4.0469 27.6875 7.5938 32.3984 Q11.1406 37.1094 17.6719 38.8125 ZM15.9219 53.7188 Q15.9219 48.6406 19.1953 45.4141 Q22.4688 42.1875 27.6875 42.1875 Q32.7656 42.1875 36.0156 45.3828 Q39.2656 48.5781 39.2656 53.2188 Q39.2656 58.0625 35.9141 61.3594 Q32.5625 64.6562 27.5938 64.6562 Q22.5625 64.6562 19.2422 61.4297 Q15.9219 58.2031 15.9219 53.7188 ZM13.0938 20.6562 Q13.0938 16.8906 14.875 13.375 Q16.6562 9.8594 20.1719 7.9297 Q23.6875 6 27.7344 6 Q34.0312 6 38.1328 10.0547 Q42.2344 14.1094 42.2344 20.3594 Q42.2344 26.7031 38.0156 30.8594 Q33.7969 35.0156 27.4375 35.0156 Q21.2344 35.0156 17.1641 30.9141 Q13.0938 26.8125 13.0938 20.6562 Z" + /><glyph unicode="9" horiz-adv-x="55.615234" d="M5.4688 16.5469 L13.9219 17.3281 Q14.9844 11.375 18.0156 8.6875 Q21.0469 6 25.7812 6 Q29.8281 6 32.8828 7.8594 Q35.9375 9.7188 37.8906 12.8203 Q39.8438 15.9219 41.1641 21.1953 Q42.4844 26.4688 42.4844 31.9375 Q42.4844 32.5156 42.4375 33.6875 Q39.7969 29.5 35.2344 26.8828 Q30.6719 24.2656 25.3438 24.2656 Q16.4531 24.2656 10.3047 30.7109 Q4.1562 37.1562 4.1562 47.7031 Q4.1562 58.5938 10.5781 65.2344 Q17 71.875 26.6562 71.875 Q33.6406 71.875 39.4297 68.1172 Q45.2188 64.3594 48.2188 57.3984 Q51.2188 50.4375 51.2188 37.25 Q51.2188 23.5312 48.2422 15.4062 Q45.2656 7.2812 39.3828 3.0312 Q33.5 -1.2188 25.5938 -1.2188 Q17.1875 -1.2188 11.8672 3.4453 Q6.5469 8.1094 5.4688 16.5469 ZM41.4531 48.1406 Q41.4531 55.7188 37.4297 60.1562 Q33.4062 64.5938 27.7344 64.5938 Q21.875 64.5938 17.5312 59.8125 Q13.1875 55.0312 13.1875 47.4062 Q13.1875 40.5781 17.3125 36.3047 Q21.4375 32.0312 27.4844 32.0312 Q33.5938 32.0312 37.5234 36.3047 Q41.4531 40.5781 41.4531 48.1406 Z" + /><glyph unicode="." horiz-adv-x="27.783203" d="M9.0781 0 L9.0781 10.0156 L19.0938 10.0156 L19.0938 0 L9.0781 0 Z" + /></font + ><font horiz-adv-x="75.0" id="font2" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:bold;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="m" horiz-adv-x="88.916016" d="M6.1562 51.8594 L18.7969 51.8594 L18.7969 44.7812 Q25.5938 53.0312 34.9688 53.0312 Q39.9375 53.0312 43.6016 50.9766 Q47.2656 48.9219 49.6094 44.7812 Q53.0312 48.9219 56.9844 50.9766 Q60.9375 53.0312 65.4375 53.0312 Q71.1406 53.0312 75.0938 50.7109 Q79.0469 48.3906 81 43.8906 Q82.4219 40.5781 82.4219 33.1562 L82.4219 0 L68.7031 0 L68.7031 29.6406 Q68.7031 37.3594 67.2812 39.5938 Q65.375 42.5312 61.4219 42.5312 Q58.5469 42.5312 56.0078 40.7734 Q53.4688 39.0156 52.3438 35.625 Q51.2188 32.2344 51.2188 24.9062 L51.2188 0 L37.5 0 L37.5 28.4219 Q37.5 35.9844 36.7656 38.1797 Q36.0312 40.375 34.4922 41.4531 Q32.9531 42.5312 30.3281 42.5312 Q27.1562 42.5312 24.6172 40.8203 Q22.0781 39.1094 20.9766 35.8906 Q19.875 32.6719 19.875 25.2031 L19.875 0 L6.1562 0 L6.1562 51.8594 Z" + /><glyph unicode="o" horiz-adv-x="61.083984" d="M4 26.6562 Q4 33.5 7.375 39.8984 Q10.75 46.2969 16.9219 49.6641 Q23.0938 53.0312 30.7188 53.0312 Q42.4844 53.0312 50 45.3906 Q57.5156 37.75 57.5156 26.0781 Q57.5156 14.3125 49.9219 6.5703 Q42.3281 -1.1719 30.8125 -1.1719 Q23.6875 -1.1719 17.2188 2.0547 Q10.75 5.2812 7.375 11.5 Q4 17.7188 4 26.6562 ZM18.0625 25.9219 Q18.0625 18.2188 21.7266 14.1172 Q25.3906 10.0156 30.7656 10.0156 Q36.1406 10.0156 39.7734 14.1172 Q43.4062 18.2188 43.4062 26.0312 Q43.4062 33.6406 39.7734 37.7422 Q36.1406 41.8438 30.7656 41.8438 Q25.3906 41.8438 21.7266 37.7422 Q18.0625 33.6406 18.0625 25.9219 Z" + /><glyph unicode="r" horiz-adv-x="38.916016" d="M20.3125 0 L6.5938 0 L6.5938 51.8594 L19.3438 51.8594 L19.3438 44.4844 Q22.6094 49.7031 25.2188 51.3672 Q27.8281 53.0312 31.1562 53.0312 Q35.8438 53.0312 40.1875 50.4375 L35.9375 38.4844 Q32.4688 40.7188 29.5 40.7188 Q26.6094 40.7188 24.6094 39.1328 Q22.6094 37.5469 21.4609 33.3984 Q20.3125 29.25 20.3125 16.0156 L20.3125 0 Z" + /><glyph unicode="d" horiz-adv-x="61.083984" d="M54.7344 0 L42 0 L42 7.625 Q38.8125 3.1719 34.4922 1 Q30.1719 -1.1719 25.7812 -1.1719 Q16.8438 -1.1719 10.4766 6.0312 Q4.1094 13.2344 4.1094 26.125 Q4.1094 39.3125 10.3047 46.1719 Q16.5 53.0312 25.9844 53.0312 Q34.6719 53.0312 41.0156 45.7969 L41.0156 71.5781 L54.7344 71.5781 L54.7344 0 ZM18.1094 27.0469 Q18.1094 18.75 20.4062 15.0469 Q23.7344 9.6719 29.6875 9.6719 Q34.4219 9.6719 37.7422 13.6953 Q41.0625 17.7188 41.0625 25.7344 Q41.0625 34.6719 37.8438 38.6016 Q34.625 42.5312 29.5938 42.5312 Q24.7031 42.5312 21.4062 38.6484 Q18.1094 34.7656 18.1094 27.0469 Z" + /><glyph unicode="n" horiz-adv-x="61.083984" d="M54.3438 0 L40.625 0 L40.625 26.4688 Q40.625 34.8594 39.75 37.3281 Q38.875 39.7969 36.8906 41.1641 Q34.9062 42.5312 32.125 42.5312 Q28.5625 42.5312 25.7344 40.5781 Q22.9062 38.625 21.8516 35.3984 Q20.7969 32.1719 20.7969 23.4844 L20.7969 0 L7.0781 0 L7.0781 51.8594 L19.8281 51.8594 L19.8281 44.2344 Q26.6094 53.0312 36.9219 53.0312 Q41.4531 53.0312 45.2109 51.3906 Q48.9688 49.75 50.8984 47.2109 Q52.8281 44.6719 53.5859 41.4531 Q54.3438 38.2344 54.3438 32.2344 L54.3438 0 Z" + /><glyph unicode="y" horiz-adv-x="55.615234" d="M0.6875 51.8594 L15.2812 51.8594 L27.6875 15.0469 L39.7969 51.8594 L54 51.8594 L35.6875 1.9531 L32.4219 -7.0781 Q30.6094 -11.625 28.9766 -14.0156 Q27.3438 -16.4062 25.2188 -17.8984 Q23.0938 -19.3906 19.9922 -20.2188 Q16.8906 -21.0469 12.9844 -21.0469 Q9.0312 -21.0469 5.2188 -20.2188 L4 -9.4688 Q7.2344 -10.1094 9.8125 -10.1094 Q14.5938 -10.1094 16.8906 -7.3047 Q19.1875 -4.5 20.4062 -0.1406 L0.6875 51.8594 Z" + /><glyph unicode="S" horiz-adv-x="66.69922" d="M3.6094 23.2969 L17.6719 24.6562 Q18.9531 17.5781 22.8281 14.2578 Q26.7031 10.9375 33.2969 10.9375 Q40.2812 10.9375 43.8203 13.8906 Q47.3594 16.8438 47.3594 20.7969 Q47.3594 23.3438 45.875 25.125 Q44.3906 26.9062 40.6719 28.2188 Q38.1406 29.1094 29.1094 31.3438 Q17.4844 34.2344 12.7969 38.4219 Q6.2031 44.3438 6.2031 52.8281 Q6.2031 58.2969 9.3047 63.0625 Q12.4062 67.8281 18.2422 70.3125 Q24.0781 72.7969 32.3281 72.7969 Q45.7969 72.7969 52.6094 66.8906 Q59.4219 60.9844 59.7656 51.125 L45.3125 50.4844 Q44.3906 56 41.3359 58.4219 Q38.2812 60.8438 32.1719 60.8438 Q25.875 60.8438 22.3125 58.25 Q20.0156 56.5938 20.0156 53.8125 Q20.0156 51.2656 22.1719 49.4688 Q24.9062 47.1719 35.4531 44.6797 Q46 42.1875 51.0547 39.5234 Q56.1094 36.8594 58.9609 32.25 Q61.8125 27.6406 61.8125 20.8438 Q61.8125 14.7031 58.3984 9.3281 Q54.9844 3.9531 48.7344 1.3438 Q42.4844 -1.2656 33.1562 -1.2656 Q19.5781 -1.2656 12.3047 5.0078 Q5.0312 11.2812 3.6094 23.2969 Z" + /></font + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >10</text + ></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >20</text + ></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >30</text + ></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >40</text + ></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >50</text + ></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >60</text + ></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >70</text + ></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >80</text + ></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >90</text + ></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="316.8333" style="fill:none;" x1="73" x2="77.34" y1="316.8333" + /><line y2="259.6667" style="fill:none;" x1="73" x2="77.34" y1="259.6667" + /><line y2="202.5" style="fill:none;" x1="73" x2="77.34" y1="202.5" + /><line y2="145.3333" style="fill:none;" x1="73" x2="77.34" y1="145.3333" + /><line y2="88.1667" style="fill:none;" x1="73" x2="77.34" y1="88.1667" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="316.8333" style="fill:none;" x1="507" x2="502.66" y1="316.8333" + /><line y2="259.6667" style="fill:none;" x1="507" x2="502.66" y1="259.6667" + /><line y2="202.5" style="fill:none;" x1="507" x2="502.66" y1="202.5" + /><line y2="145.3333" style="fill:none;" x1="507" x2="502.66" y1="145.3333" + /><line y2="88.1667" style="fill:none;" x1="507" x2="502.66" y1="88.1667" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >0</text + ></g + ><g transform="translate(67.6667,316.8333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-26" xml:space="preserve" y="5.5" style="stroke:none;" + >0.01</text + ></g + ><g transform="translate(67.6667,259.6667)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-26" xml:space="preserve" y="5.5" style="stroke:none;" + >0.02</text + ></g + ><g transform="translate(67.6667,202.5)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-26" xml:space="preserve" y="5.5" style="stroke:none;" + >0.03</text + ></g + ><g transform="translate(67.6667,145.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-26" xml:space="preserve" y="5.5" style="stroke:none;" + >0.04</text + ></g + ><g transform="translate(67.6667,88.1667)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-26" xml:space="preserve" y="5.5" style="stroke:none;" + >0.05</text + ></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-26" xml:space="preserve" y="5.5" style="stroke:none;" + >0.06</text + ></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><text x="-32" xml:space="preserve" y="-4" style="stroke:none;" + >Syndrom</text + ></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 374 L77.5684 374 L82.1368 374 L86.7053 374 L91.2737 374 L95.8421 374 L100.4105 374 L104.979 374 L109.5474 374 L114.1158 374 L118.6842 374 L123.2526 374 L127.8211 374 L132.3895 374 L136.9579 374 L141.5263 374 L146.0947 374 L150.6632 374 L155.2316 374 L159.8 374 L164.3684 374 L168.9368 374 L173.5053 374 L178.0737 374 L182.6421 374 L187.2105 374 L191.7789 374 L196.3474 374 L200.9158 374 L205.4842 374 L210.0526 374 L214.621 374 L219.1895 374 L223.7579 374 L228.3263 374 L232.8947 374 L237.4632 374 L242.0316 374 L246.6 374 L251.1684 374 L255.7368 374 L260.3053 374 L264.8737 374 L269.4421 374 L274.0105 374 L278.5789 374 L283.1474 374 L287.7158 374 L292.2842 374 L296.8526 374 L301.4211 374 L305.9895 374 L310.5579 374 L315.1263 374 L319.6947 374 L324.2632 374 L328.8316 374 L333.4 374 L337.9684 374 L342.5368 374 L347.1053 374 L351.6737 374 L356.2421 374 L360.8105 374 L365.3789 216.4492 L369.9474 307.7798 L374.5158 226.0655 L379.0842 245.5934 L383.6526 121.861 L388.221 102.7419 L392.7895 243.8345 L397.3579 218.8287 L401.9263 314.4514 L406.4948 218.8287 L411.0632 243.8345 L415.6316 102.7419 L420.2 121.861 L424.7684 245.5934 L429.3369 226.0655 L433.9053 307.7798 L438.4737 216.4492 L443.0421 233.8706 L447.6105 88.2912 L452.179 144.5147 L456.7474 240.8453 L461.3158 237.958 L465.8842 291.7395 L470.4526 218.652 L475.0211 216.4492 L479.5895 79.3049 L484.1579 169.1975 L488.7263 232.7954 L493.2947 253.8387 L497.8632 272.476 L502.4316 224.6081 L507 193.977" style="fill:none; fill-rule:evenodd;" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/fig7.pdf b/buch/papers/reedsolomon/RS presentation/images/fig7.pdf Binary files differnew file mode 100644 index 0000000..7397ffe --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig7.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/fig7.svg b/buch/papers/reedsolomon/RS presentation/images/fig7.svg new file mode 100644 index 0000000..27c7622 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/fig7.svg @@ -0,0 +1,163 @@ +<?xml version="1.0"?> +<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.0//EN' + 'http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd'> +<svg xmlns:xlink="http://www.w3.org/1999/xlink" style="fill-opacity:1; color-rendering:auto; color-interpolation:auto; text-rendering:auto; stroke:black; stroke-linecap:square; stroke-miterlimit:10; shape-rendering:auto; stroke-opacity:1; fill:black; stroke-dasharray:none; font-weight:normal; stroke-width:1; font-family:'Dialog'; font-style:normal; stroke-linejoin:miter; font-size:12px; stroke-dashoffset:0; image-rendering:auto;" width="560" height="420" xmlns="http://www.w3.org/2000/svg" +><!--Generated by the Batik Graphics2D SVG Generator--><defs id="genericDefs" + /><g + ><defs id="defs1" + ><clipPath clipPathUnits="userSpaceOnUse" id="clipPath1" + ><path d="M0 0 L560 0 L560 420 L0 420 L0 0 Z" + /></clipPath + ><font horiz-adv-x="75.0" id="font1" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:normal;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="0" horiz-adv-x="55.615234" d="M4.1562 35.2969 Q4.1562 48 6.7656 55.7422 Q9.375 63.4844 14.5234 67.6797 Q19.6719 71.875 27.4844 71.875 Q33.25 71.875 37.5938 69.5547 Q41.9375 67.2344 44.7734 62.8672 Q47.6094 58.5 49.2188 52.2266 Q50.8281 45.9531 50.8281 35.2969 Q50.8281 22.7031 48.2422 14.9688 Q45.6562 7.2344 40.5078 3.0078 Q35.3594 -1.2188 27.4844 -1.2188 Q17.1406 -1.2188 11.2344 6.2031 Q4.1562 15.1406 4.1562 35.2969 ZM13.1875 35.2969 Q13.1875 17.6719 17.3125 11.8359 Q21.4375 6 27.4844 6 Q33.5469 6 37.6719 11.8594 Q41.7969 17.7188 41.7969 35.2969 Q41.7969 52.9844 37.6719 58.7891 Q33.5469 64.5938 27.3906 64.5938 Q21.3438 64.5938 17.7188 59.4688 Q13.1875 52.9375 13.1875 35.2969 Z" + /><glyph unicode="1" horiz-adv-x="55.615234" d="M37.25 0 L28.4688 0 L28.4688 56 Q25.2969 52.9844 20.1406 49.9531 Q14.9844 46.9219 10.8906 45.4062 L10.8906 53.9062 Q18.2656 57.375 23.7812 62.3047 Q29.2969 67.2344 31.5938 71.875 L37.25 71.875 L37.25 0 Z" + /><glyph unicode="2" horiz-adv-x="55.615234" d="M50.3438 8.4531 L50.3438 0 L3.0312 0 Q2.9375 3.1719 4.0469 6.1094 Q5.8594 10.9375 9.8359 15.625 Q13.8125 20.3125 21.3438 26.4688 Q33.0156 36.0312 37.1172 41.625 Q41.2188 47.2188 41.2188 52.2031 Q41.2188 57.4219 37.4766 61.0078 Q33.7344 64.5938 27.7344 64.5938 Q21.3906 64.5938 17.5781 60.7891 Q13.7656 56.9844 13.7188 50.25 L4.6875 51.1719 Q5.6094 61.2812 11.6641 66.5781 Q17.7188 71.875 27.9375 71.875 Q38.2344 71.875 44.2422 66.1641 Q50.25 60.4531 50.25 52 Q50.25 47.7031 48.4922 43.5547 Q46.7344 39.4062 42.6562 34.8125 Q38.5781 30.2188 29.1094 22.2188 Q21.1875 15.5781 18.9453 13.2109 Q16.7031 10.8438 15.2344 8.4531 L50.3438 8.4531 Z" + /><glyph unicode="3" horiz-adv-x="55.615234" d="M4.2031 18.8906 L12.9844 20.0625 Q14.5 12.5938 18.1406 9.2969 Q21.7812 6 27 6 Q33.2031 6 37.4766 10.2969 Q41.75 14.5938 41.75 20.9531 Q41.75 27 37.7969 30.9297 Q33.8438 34.8594 27.7344 34.8594 Q25.25 34.8594 21.5312 33.8906 L22.5156 41.6094 Q23.3906 41.5 23.9219 41.5 Q29.5469 41.5 34.0391 44.4297 Q38.5312 47.3594 38.5312 53.4688 Q38.5312 58.2969 35.2578 61.4766 Q31.9844 64.6562 26.8125 64.6562 Q21.6875 64.6562 18.2656 61.4297 Q14.8438 58.2031 13.875 51.7656 L5.0781 53.3281 Q6.6875 62.1562 12.3984 67.0156 Q18.1094 71.875 26.6094 71.875 Q32.4688 71.875 37.3984 69.3594 Q42.3281 66.8438 44.9453 62.5 Q47.5625 58.1562 47.5625 53.2656 Q47.5625 48.6406 45.0703 44.8281 Q42.5781 41.0156 37.7031 38.7656 Q44.0469 37.3125 47.5625 32.6953 Q51.0781 28.0781 51.0781 21.1406 Q51.0781 11.7656 44.2422 5.25 Q37.4062 -1.2656 26.9531 -1.2656 Q17.5312 -1.2656 11.3047 4.3516 Q5.0781 9.9688 4.2031 18.8906 Z" + /><glyph unicode="4" horiz-adv-x="55.615234" d="M32.3281 0 L32.3281 17.1406 L1.2656 17.1406 L1.2656 25.2031 L33.9375 71.5781 L41.1094 71.5781 L41.1094 25.2031 L50.7812 25.2031 L50.7812 17.1406 L41.1094 17.1406 L41.1094 0 L32.3281 0 ZM32.3281 25.2031 L32.3281 57.4688 L9.9062 25.2031 L32.3281 25.2031 Z" + /><glyph unicode="5" horiz-adv-x="55.615234" d="M4.1562 18.75 L13.375 19.5312 Q14.4062 12.7969 18.1406 9.3984 Q21.875 6 27.1562 6 Q33.5 6 37.8906 10.7891 Q42.2812 15.5781 42.2812 23.4844 Q42.2812 31 38.0625 35.3516 Q33.8438 39.7031 27 39.7031 Q22.75 39.7031 19.3359 37.7734 Q15.9219 35.8438 13.9688 32.7656 L5.7188 33.8438 L12.6406 70.6094 L48.25 70.6094 L48.25 62.2031 L19.6719 62.2031 L15.8281 42.9688 Q22.2656 47.4688 29.3438 47.4688 Q38.7188 47.4688 45.1641 40.9688 Q51.6094 34.4688 51.6094 24.2656 Q51.6094 14.5469 45.9531 7.4688 Q39.0625 -1.2188 27.1562 -1.2188 Q17.3906 -1.2188 11.2109 4.25 Q5.0312 9.7188 4.1562 18.75 Z" + /><glyph unicode="6" horiz-adv-x="55.615234" d="M49.75 54.0469 L41.0156 53.375 Q39.8438 58.5469 37.7031 60.8906 Q34.125 64.6562 28.9062 64.6562 Q24.7031 64.6562 21.5312 62.3125 Q17.3906 59.2812 14.9922 53.4688 Q12.5938 47.6562 12.5 36.9219 Q15.6719 41.75 20.2656 44.0938 Q24.8594 46.4375 29.8906 46.4375 Q38.6719 46.4375 44.8516 39.9688 Q51.0312 33.5 51.0312 23.25 Q51.0312 16.5 48.125 10.7188 Q45.2188 4.9375 40.1406 1.8594 Q35.0625 -1.2188 28.6094 -1.2188 Q17.625 -1.2188 10.6953 6.8594 Q3.7656 14.9375 3.7656 33.5 Q3.7656 54.25 11.4219 63.6719 Q18.1094 71.875 29.4375 71.875 Q37.8906 71.875 43.2891 67.1406 Q48.6875 62.4062 49.75 54.0469 ZM13.875 23.1875 Q13.875 18.6562 15.7969 14.5078 Q17.7188 10.3594 21.1875 8.1797 Q24.6562 6 28.4688 6 Q34.0312 6 38.0391 10.4922 Q42.0469 14.9844 42.0469 22.7031 Q42.0469 30.125 38.0859 34.3984 Q34.125 38.6719 28.125 38.6719 Q22.1719 38.6719 18.0234 34.3984 Q13.875 30.125 13.875 23.1875 Z" + /><glyph unicode="7" horiz-adv-x="55.615234" d="M4.7344 62.2031 L4.7344 70.6562 L51.0781 70.6562 L51.0781 63.8125 Q44.2344 56.5469 37.5234 44.4844 Q30.8125 32.4219 27.1562 19.6719 Q24.5156 10.6875 23.7812 0 L14.75 0 Q14.8906 8.4531 18.0625 20.4141 Q21.2344 32.375 27.1719 43.4844 Q33.1094 54.5938 39.7969 62.2031 L4.7344 62.2031 Z" + /><glyph unicode="8" horiz-adv-x="55.615234" d="M17.6719 38.8125 Q12.2031 40.8281 9.5703 44.5391 Q6.9375 48.25 6.9375 53.4219 Q6.9375 61.2344 12.5547 66.5547 Q18.1719 71.875 27.4844 71.875 Q36.8594 71.875 42.5781 66.4297 Q48.2969 60.9844 48.2969 53.1719 Q48.2969 48.1875 45.6797 44.5078 Q43.0625 40.8281 37.75 38.8125 Q44.3438 36.6719 47.7812 31.8828 Q51.2188 27.0938 51.2188 20.4531 Q51.2188 11.2812 44.7266 5.0312 Q38.2344 -1.2188 27.6406 -1.2188 Q17.0469 -1.2188 10.5469 5.0547 Q4.0469 11.3281 4.0469 20.7031 Q4.0469 27.6875 7.5938 32.3984 Q11.1406 37.1094 17.6719 38.8125 ZM15.9219 53.7188 Q15.9219 48.6406 19.1953 45.4141 Q22.4688 42.1875 27.6875 42.1875 Q32.7656 42.1875 36.0156 45.3828 Q39.2656 48.5781 39.2656 53.2188 Q39.2656 58.0625 35.9141 61.3594 Q32.5625 64.6562 27.5938 64.6562 Q22.5625 64.6562 19.2422 61.4297 Q15.9219 58.2031 15.9219 53.7188 ZM13.0938 20.6562 Q13.0938 16.8906 14.875 13.375 Q16.6562 9.8594 20.1719 7.9297 Q23.6875 6 27.7344 6 Q34.0312 6 38.1328 10.0547 Q42.2344 14.1094 42.2344 20.3594 Q42.2344 26.7031 38.0156 30.8594 Q33.7969 35.0156 27.4375 35.0156 Q21.2344 35.0156 17.1641 30.9141 Q13.0938 26.8125 13.0938 20.6562 Z" + /><glyph unicode="9" horiz-adv-x="55.615234" d="M5.4688 16.5469 L13.9219 17.3281 Q14.9844 11.375 18.0156 8.6875 Q21.0469 6 25.7812 6 Q29.8281 6 32.8828 7.8594 Q35.9375 9.7188 37.8906 12.8203 Q39.8438 15.9219 41.1641 21.1953 Q42.4844 26.4688 42.4844 31.9375 Q42.4844 32.5156 42.4375 33.6875 Q39.7969 29.5 35.2344 26.8828 Q30.6719 24.2656 25.3438 24.2656 Q16.4531 24.2656 10.3047 30.7109 Q4.1562 37.1562 4.1562 47.7031 Q4.1562 58.5938 10.5781 65.2344 Q17 71.875 26.6562 71.875 Q33.6406 71.875 39.4297 68.1172 Q45.2188 64.3594 48.2188 57.3984 Q51.2188 50.4375 51.2188 37.25 Q51.2188 23.5312 48.2422 15.4062 Q45.2656 7.2812 39.3828 3.0312 Q33.5 -1.2188 25.5938 -1.2188 Q17.1875 -1.2188 11.8672 3.4453 Q6.5469 8.1094 5.4688 16.5469 ZM41.4531 48.1406 Q41.4531 55.7188 37.4297 60.1562 Q33.4062 64.5938 27.7344 64.5938 Q21.875 64.5938 17.5312 59.8125 Q13.1875 55.0312 13.1875 47.4062 Q13.1875 40.5781 17.3125 36.3047 Q21.4375 32.0312 27.4844 32.0312 Q33.5938 32.0312 37.5234 36.3047 Q41.4531 40.5781 41.4531 48.1406 Z" + /><glyph unicode="." horiz-adv-x="27.783203" d="M9.0781 0 L9.0781 10.0156 L19.0938 10.0156 L19.0938 0 L9.0781 0 Z" + /></font + ><font horiz-adv-x="75.0" id="font2" + ><font-face ascent="100.53711" descent="21.972656" units-per-em="100" style="font-style:normal; font-family:SansSerif; font-weight:bold;" + /><missing-glyph horiz-adv-x="75.0" d="M12.5 0 L12.5 62.5 L62.5 62.5 L62.5 0 L12.5 0 ZM14.0625 1.5625 L60.9375 1.5625 L60.9375 60.9375 L14.0625 60.9375 L14.0625 1.5625 Z" + /><glyph unicode="r" horiz-adv-x="38.916016" d="M20.3125 0 L6.5938 0 L6.5938 51.8594 L19.3438 51.8594 L19.3438 44.4844 Q22.6094 49.7031 25.2188 51.3672 Q27.8281 53.0312 31.1562 53.0312 Q35.8438 53.0312 40.1875 50.4375 L35.9375 38.4844 Q32.4688 40.7188 29.5 40.7188 Q26.6094 40.7188 24.6094 39.1328 Q22.6094 37.5469 21.4609 33.3984 Q20.3125 29.25 20.3125 16.0156 L20.3125 0 Z" + /><glyph unicode="t" horiz-adv-x="33.30078" d="M30.9531 51.8594 L30.9531 40.9219 L21.5781 40.9219 L21.5781 20.0156 Q21.5781 13.6719 21.8516 12.625 Q22.125 11.5781 23.0781 10.8906 Q24.0312 10.2031 25.3906 10.2031 Q27.2969 10.2031 30.9062 11.5312 L32.0781 0.875 Q27.2969 -1.1719 21.2344 -1.1719 Q17.5312 -1.1719 14.5547 0.0703 Q11.5781 1.3125 10.1875 3.2969 Q8.7969 5.2812 8.25 8.6406 Q7.8125 11.0312 7.8125 18.3125 L7.8125 40.9219 L1.5156 40.9219 L1.5156 51.8594 L7.8125 51.8594 L7.8125 62.1562 L21.5781 70.1719 L21.5781 51.8594 L30.9531 51.8594 Z" + /><glyph unicode="a" horiz-adv-x="55.615234" d="M17.4375 36.0312 L4.9844 38.2812 Q7.0781 45.7969 12.2031 49.4141 Q17.3281 53.0312 27.4375 53.0312 Q36.625 53.0312 41.1172 50.8594 Q45.6094 48.6875 47.4375 45.3438 Q49.2656 42 49.2656 33.0625 L49.125 17.0469 Q49.125 10.2031 49.7812 6.9609 Q50.4375 3.7188 52.25 0 L38.6719 0 Q38.1406 1.375 37.3594 4.0469 Q37.0156 5.2812 36.8594 5.6719 Q33.3438 2.25 29.3438 0.5391 Q25.3438 -1.1719 20.7969 -1.1719 Q12.7969 -1.1719 8.1797 3.1719 Q3.5625 7.5156 3.5625 14.1562 Q3.5625 18.5625 5.6641 22 Q7.7656 25.4375 11.5469 27.2734 Q15.3281 29.1094 22.4688 30.4688 Q32.0781 32.2812 35.7969 33.8438 L35.7969 35.2031 Q35.7969 39.1562 33.8438 40.8438 Q31.8906 42.5312 26.4688 42.5312 Q22.7969 42.5312 20.75 41.0938 Q18.7031 39.6562 17.4375 36.0312 ZM35.7969 24.9062 Q33.1562 24.0312 27.4453 22.8047 Q21.7344 21.5781 19.9688 20.4062 Q17.2812 18.5 17.2812 15.5781 Q17.2812 12.7031 19.4297 10.6016 Q21.5781 8.5 24.9062 8.5 Q28.6094 8.5 31.9844 10.9375 Q34.4688 12.7969 35.25 15.4844 Q35.7969 17.2344 35.7969 22.1719 L35.7969 24.9062 Z" + /><glyph unicode="c" horiz-adv-x="55.615234" d="M52.3906 36.5312 L38.875 34.0781 Q38.1875 38.1406 35.7656 40.1875 Q33.3438 42.2344 29.5 42.2344 Q24.3594 42.2344 21.3125 38.6953 Q18.2656 35.1562 18.2656 26.8594 Q18.2656 17.625 21.3672 13.8203 Q24.4688 10.0156 29.6875 10.0156 Q33.5938 10.0156 36.0859 12.2344 Q38.5781 14.4531 39.5938 19.875 L53.0781 17.5781 Q50.9844 8.2969 45.0234 3.5625 Q39.0625 -1.1719 29.0469 -1.1719 Q17.6719 -1.1719 10.9141 6.0078 Q4.1562 13.1875 4.1562 25.875 Q4.1562 38.7188 10.9375 45.875 Q17.7188 53.0312 29.2969 53.0312 Q38.7656 53.0312 44.3594 48.9531 Q49.9531 44.875 52.3906 36.5312 Z" + /><glyph unicode="o" horiz-adv-x="61.083984" d="M4 26.6562 Q4 33.5 7.375 39.8984 Q10.75 46.2969 16.9219 49.6641 Q23.0938 53.0312 30.7188 53.0312 Q42.4844 53.0312 50 45.3906 Q57.5156 37.75 57.5156 26.0781 Q57.5156 14.3125 49.9219 6.5703 Q42.3281 -1.1719 30.8125 -1.1719 Q23.6875 -1.1719 17.2188 2.0547 Q10.75 5.2812 7.375 11.5 Q4 17.7188 4 26.6562 ZM18.0625 25.9219 Q18.0625 18.2188 21.7266 14.1172 Q25.3906 10.0156 30.7656 10.0156 Q36.1406 10.0156 39.7734 14.1172 Q43.4062 18.2188 43.4062 26.0312 Q43.4062 33.6406 39.7734 37.7422 Q36.1406 41.8438 30.7656 41.8438 Q25.3906 41.8438 21.7266 37.7422 Q18.0625 33.6406 18.0625 25.9219 Z" + /><glyph unicode="L" horiz-adv-x="61.083984" d="M7.6719 0 L7.6719 71 L22.125 71 L22.125 12.0625 L58.0625 12.0625 L58.0625 0 L7.6719 0 Z" + /></font + ></defs + ><g style="fill:white; stroke:white;" + ><rect x="0" y="0" width="560" style="clip-path:url(#clipPath1); stroke:none;" height="420" + /></g + ><g style="fill:white; text-rendering:optimizeSpeed; color-rendering:optimizeSpeed; image-rendering:optimizeSpeed; shape-rendering:crispEdges; stroke:white; color-interpolation:sRGB;" + ><rect x="0" width="560" height="420" y="0" style="stroke:none;" + /><path style="stroke:none;" d="M73 374 L507 374 L507 31 L73 31 Z" + /></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="374" style="fill:none;" x1="73" x2="507" y1="374" + /><line y2="31" style="fill:none;" x1="73" x2="507" y1="31" + /><line y2="369.66" style="fill:none;" x1="114.1158" x2="114.1158" y1="374" + /><line y2="369.66" style="fill:none;" x1="159.8" x2="159.8" y1="374" + /><line y2="369.66" style="fill:none;" x1="205.4842" x2="205.4842" y1="374" + /><line y2="369.66" style="fill:none;" x1="251.1684" x2="251.1684" y1="374" + /><line y2="369.66" style="fill:none;" x1="296.8526" x2="296.8526" y1="374" + /><line y2="369.66" style="fill:none;" x1="342.5368" x2="342.5368" y1="374" + /><line y2="369.66" style="fill:none;" x1="388.221" x2="388.221" y1="374" + /><line y2="369.66" style="fill:none;" x1="433.9053" x2="433.9053" y1="374" + /><line y2="369.66" style="fill:none;" x1="479.5895" x2="479.5895" y1="374" + /><line y2="35.34" style="fill:none;" x1="114.1158" x2="114.1158" y1="31" + /><line y2="35.34" style="fill:none;" x1="159.8" x2="159.8" y1="31" + /><line y2="35.34" style="fill:none;" x1="205.4842" x2="205.4842" y1="31" + /><line y2="35.34" style="fill:none;" x1="251.1684" x2="251.1684" y1="31" + /><line y2="35.34" style="fill:none;" x1="296.8526" x2="296.8526" y1="31" + /><line y2="35.34" style="fill:none;" x1="342.5368" x2="342.5368" y1="31" + /><line y2="35.34" style="fill:none;" x1="388.221" x2="388.221" y1="31" + /><line y2="35.34" style="fill:none;" x1="433.9053" x2="433.9053" y1="31" + /><line y2="35.34" style="fill:none;" x1="479.5895" x2="479.5895" y1="31" + /></g + ><g transform="translate(114.1158,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >10</text + ></g + ><g transform="translate(159.8,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >20</text + ></g + ><g transform="translate(205.4842,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >30</text + ></g + ><g transform="translate(251.1684,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >40</text + ></g + ><g transform="translate(296.8526,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >50</text + ></g + ><g transform="translate(342.5368,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >60</text + ></g + ><g transform="translate(388.221,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >70</text + ></g + ><g transform="translate(433.9053,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >80</text + ></g + ><g transform="translate(479.5895,379.3333)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;" + >90</text + ></g + ><g style="fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(38,38,38); color-interpolation:linearRGB; stroke-width:0.6667;" + ><line y2="31" style="fill:none;" x1="73" x2="73" y1="374" + /><line y2="31" style="fill:none;" x1="507" x2="507" y1="374" + /><line y2="374" style="fill:none;" x1="73" x2="77.34" y1="374" + /><line y2="325" style="fill:none;" x1="73" x2="77.34" y1="325" + /><line y2="276" style="fill:none;" x1="73" x2="77.34" y1="276" + /><line y2="227" style="fill:none;" x1="73" x2="77.34" y1="227" + /><line y2="178" style="fill:none;" x1="73" x2="77.34" y1="178" + /><line y2="129" style="fill:none;" x1="73" x2="77.34" y1="129" + /><line y2="80" style="fill:none;" x1="73" x2="77.34" y1="80" + /><line y2="31" style="fill:none;" x1="73" x2="77.34" y1="31" + /><line y2="374" style="fill:none;" x1="507" x2="502.66" y1="374" + /><line y2="325" style="fill:none;" x1="507" x2="502.66" y1="325" + /><line y2="276" style="fill:none;" x1="507" x2="502.66" y1="276" + /><line y2="227" style="fill:none;" x1="507" x2="502.66" y1="227" + /><line y2="178" style="fill:none;" x1="507" x2="502.66" y1="178" + /><line y2="129" style="fill:none;" x1="507" x2="502.66" y1="129" + /><line y2="80" style="fill:none;" x1="507" x2="502.66" y1="80" + /><line y2="31" style="fill:none;" x1="507" x2="502.66" y1="31" + /></g + ><g transform="translate(67.6667,374)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;" + >0</text + ></g + ><g transform="translate(67.6667,325)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.1</text + ></g + ><g transform="translate(67.6667,276)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.2</text + ></g + ><g transform="translate(67.6667,227)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.3</text + ></g + ><g transform="translate(67.6667,178)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.4</text + ></g + ><g transform="translate(67.6667,129)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.5</text + ></g + ><g transform="translate(67.6667,80)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.6</text + ></g + ><g transform="translate(67.6667,31)" style="font-size:13.3333px; fill:rgb(38,38,38); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; stroke:rgb(38,38,38); color-interpolation:linearRGB;" + ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;" + >0.7</text + ></g + ><g transform="translate(290.0003,28.25)" style="font-size:14.6667px; text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; font-family:'SansSerif'; color-interpolation:linearRGB; font-weight:bold;" + ><text x="-27" xml:space="preserve" y="-4" style="stroke:none;" + >Locator</text + ></g + ><g style="stroke-linecap:butt; fill:rgb(0,114,189); text-rendering:geometricPrecision; image-rendering:optimizeQuality; color-rendering:optimizeQuality; stroke-linejoin:round; stroke:rgb(0,114,189); color-interpolation:linearRGB; stroke-width:0.6667;" + ><path d="M73 359.24 L77.5684 304.59 L82.1368 306.269 L86.7053 357.3447 L91.2737 221.8131 L95.8421 103.8006 L100.4105 66.0277 L104.979 123.5983 L109.5474 238.8279 L114.1158 349.3826 L118.6842 329.8091 L123.2526 319.7279 L127.8211 340.4537 L132.3895 356.2201 L136.9579 339.8423 L141.5263 320.4486 L146.0947 328.7638 L150.6632 348.9023 L155.2316 239.6457 L159.8 122.3869 L164.3684 63.7543 L168.9368 102.891 L173.5053 223.1581 L178.0737 357.2585 L182.6421 305.2673 L187.2105 306.1217 L191.7789 359.4776 L196.3474 333.9688 L200.9158 327.7252 L205.4842 361.0673 L210.0526 345.2918 L214.621 337.8813 L219.1895 362.2413 L223.7579 351.2534 L228.3263 343.8053 L232.8947 363.1238 L237.4632 354.8525 L242.0316 347.6558 L246.6 363.7917 L251.1684 357.1877 L255.7368 350.3114 L260.3053 364.2942 L264.8737 358.7508 L269.4421 352.1972 L274.0105 364.6638 L278.5789 359.7876 L283.1474 353.5407 L287.7158 364.9222 L292.2842 360.4241 L296.8526 354.4683 L301.4211 365.083 L305.9895 360.7162 L310.5579 355.045 L315.1263 365.1546 L319.6947 360.6652 L324.2632 355.2873 L328.8316 365.1404 L333.4 360.2105 L337.9684 355.1548 L342.5368 365.0396 L347.1053 359.1897 L351.6737 354.5041 L356.2421 364.8475 L360.8105 357.218 L365.3789 352.9333 L369.9474 364.5539 L374.5158 353.2823 L379.0842 349.166 L383.6526 364.1428 L388.221 343.8626 L392.7895 337.4973 L397.3579 363.5894 L401.9263 304.1249 L406.4948 239.984 L411.0632 214.5983 L415.6316 240.3744 L420.2 300.9536 L424.7684 361.8872 L429.3369 340.6732 L433.9053 337.0598 L438.4737 360.5917 L443.0421 354.0145 L447.6105 342.9847 L452.179 358.8223 L456.7474 358.5551 L461.3158 343.2464 L465.8842 356.3186 L470.4526 360.3049 L475.0211 340.7819 L479.5895 352.5788 L484.1579 360.7386 L488.7263 335.6144 L493.2947 346.4966 L497.8632 360.3416 L502.4316 325.9092 L507 335.0674" style="fill:none; fill-rule:evenodd;" + /></g + ></g +></svg +> diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.aux b/buch/papers/reedsolomon/RS presentation/images/polynom1.aux new file mode 100644 index 0000000..f23e546 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.aux @@ -0,0 +1 @@ +\relax diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.log b/buch/papers/reedsolomon/RS presentation/images/polynom1.log new file mode 100644 index 0000000..183eaea --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.log @@ -0,0 +1,747 @@ +This is XeTeX, Version 3.14159265-2.6-0.999991 (TeX Live 2019/W32TeX) (preloaded format=xelatex 2019.10.25) 21 APR 2021 12:27 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**polynom1.tex +(./polynom1.tex +LaTeX2e <2019-10-01> patch level 1 +(c:/texlive/2019/texmf-dist/tex/latex/standalone/standalone.cls +Document Class: standalone 2018/03/26 v1.3a Class to compile TeX sub-files stan +dalone +(c:/texlive/2019/texmf-dist/tex/latex/tools/shellesc.sty +Package: shellesc 2016/06/07 v0.02b unified shell escape interface for LaTeX +Package shellesc Info: Restricted shell escape enabled on input line 72. +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifluatex.sty +Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch +) +(c:/texlive/2019/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +(c:/texlive/2019/texmf-dist/tex/latex/xkeyval/xkeyval.sty +Package: xkeyval 2014/12/03 v2.7a package option processing (HA) + +(c:/texlive/2019/texmf-dist/tex/generic/xkeyval/xkeyval.tex +(c:/texlive/2019/texmf-dist/tex/generic/xkeyval/xkvutils.tex +\XKV@toks=\toks14 +\XKV@tempa@toks=\toks15 + +(c:/texlive/2019/texmf-dist/tex/generic/xkeyval/keyval.tex)) +\XKV@depth=\count80 +File: xkeyval.tex 2014/12/03 v2.7a key=value parser (HA) +)) +\sa@internal=\count81 +\c@sapage=\count82 + +(c:/texlive/2019/texmf-dist/tex/latex/standalone/standalone.cfg +File: standalone.cfg 2018/03/26 v1.3a Default configuration file for 'standalon +e' class +) +(c:/texlive/2019/texmf-dist/tex/latex/base/article.cls +Document Class: article 2019/08/27 v1.4j Standard LaTeX document class +(c:/texlive/2019/texmf-dist/tex/latex/base/size10.clo +File: size10.clo 2019/08/27 v1.4j Standard LaTeX file (size option) +) +\c@part=\count83 +\c@section=\count84 +\c@subsection=\count85 +\c@subsubsection=\count86 +\c@paragraph=\count87 +\c@subparagraph=\count88 +\c@figure=\count89 +\c@table=\count90 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks16 +\pgfutil@tempdima=\dimen103 +\pgfutil@tempdimb=\dimen104 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box27 + +(c:/texlive/2019/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/pgf.revision.tex) +Package: pgfrcs 2019/08/03 v3.1.4b (3.1.4b) +)) +Package: pgf 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2019/10/08 v1.3c Standard LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 2016/01/03 v1.10 sin cos tan (DPC) +) +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration +) +Package graphics Info: Driver file: xetex.def on input line 105. + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-def/xetex.def +File: xetex.def 2017/06/24 v5.0h Graphics/color driver for xetex +)) +\Gin@req@height=\dimen105 +\Gin@req@width=\dimen106 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks17 +\pgfkeys@temptoks=\toks18 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks19 +)) +\pgf@x=\dimen107 +\pgf@y=\dimen108 +\pgf@xa=\dimen109 +\pgf@ya=\dimen110 +\pgf@xb=\dimen111 +\pgf@yb=\dimen112 +\pgf@xc=\dimen113 +\pgf@yc=\dimen114 +\pgf@xd=\dimen115 +\pgf@yd=\dimen116 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count91 +\c@pgf@countb=\count92 +\c@pgf@countc=\count93 +\c@pgf@countd=\count94 +\t@pgf@toka=\toks20 +\t@pgf@tokb=\toks21 +\t@pgf@tokc=\toks22 +\pgf@sys@id@count=\count95 + (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2019/08/03 v3.1.4b (3.1.4b) +) +Driver file for pgf: pgfsys-xetex.def + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-xetex.def +File: pgfsys-xetex.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-dvipdfmx.def +File: pgfsys-dvipdfmx.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2019/08/03 v3.1.4b (3.1.4b) +) +\pgfsys@objnum=\count96 +))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfsyssoftpath@smallbuffer@items=\count97 +\pgfsyssoftpath@bigbuffer@items=\count98 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/xcolor/xcolor.sty +Package: xcolor 2016/05/11 v2.12 LaTeX color extensions (UK) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/color.cfg +File: color.cfg 2016/01/02 v1.6 sample color configuration +) +Package xcolor Info: Driver file: xetex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1348. +Package xcolor Info: Model `RGB' extended on input line 1364. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1366. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1367. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1368. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1369. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1370. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1371. +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen117 +\pgfmath@count=\count99 +\pgfmath@box=\box28 +\pgfmath@toks=\toks23 +\pgfmath@stack@operand=\toks24 +\pgfmath@stack@operation=\toks25 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric +.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.t +ex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.co +de.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithm +etics.code.tex))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count100 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@picminx=\dimen118 +\pgf@picmaxx=\dimen119 +\pgf@picminy=\dimen120 +\pgf@picmaxy=\dimen121 +\pgf@pathminx=\dimen122 +\pgf@pathmaxx=\dimen123 +\pgf@pathminy=\dimen124 +\pgf@pathmaxy=\dimen125 +\pgf@xx=\dimen126 +\pgf@xy=\dimen127 +\pgf@yx=\dimen128 +\pgf@yy=\dimen129 +\pgf@zx=\dimen130 +\pgf@zy=\dimen131 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.cod +e.tex +File: pgfcorepathconstruct.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@path@lastx=\dimen132 +\pgf@path@lasty=\dimen133 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.te +x +File: pgfcorepathusage.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@shorten@end@additional=\dimen134 +\pgf@shorten@start@additional=\dimen135 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfpic=\box29 +\pgf@hbox=\box30 +\pgf@layerbox@main=\box31 +\pgf@picture@serial@count=\count101 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code +.tex +File: pgfcoregraphicstate.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgflinewidth=\dimen136 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.c +ode.tex +File: pgfcoretransformations.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@pt@x=\dimen137 +\pgf@pt@y=\dimen138 +\pgf@pt@temp=\dimen139 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.co +de.tex +File: pgfcorepathprocessing.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfarrowsep=\dimen140 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@max=\dimen141 +\pgf@sys@shading@range@num=\count102 +\pgf@shadingcount=\count103 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfexternal@startupbox=\box32 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code +.tex +File: pgfcoretransparency.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +File: pgfcorerdf.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) (c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.te +x +File: pgfmoduleshapes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfnodeparttextbox=\box33 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.st +y +Package: pgfcomp-version-0-65 2019/08/03 v3.1.4b (3.1.4b) +\pgf@nodesepstart=\dimen142 +\pgf@nodesepend=\dimen143 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.st +y +Package: pgfcomp-version-1-18 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/math/pgfmath.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex) +\pgffor@iter=\dimen144 +\pgffor@skip=\dimen145 +\pgffor@stack=\toks26 +\pgffor@toks=\toks27 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.co +de.tex +File: pgflibraryplothandlers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@plot@mark@count=\count104 +\pgfplotmarksize=\dimen146 +) +\tikz@lastx=\dimen147 +\tikz@lasty=\dimen148 +\tikz@lastxsaved=\dimen149 +\tikz@lastysaved=\dimen150 +\tikz@lastmovetox=\dimen151 +\tikz@lastmovetoy=\dimen152 +\tikzleveldistance=\dimen153 +\tikzsiblingdistance=\dimen154 +\tikz@figbox=\box34 +\tikz@figbox@bg=\box35 +\tikz@tempbox=\box36 +\tikz@tempbox@bg=\box37 +\tikztreelevel=\count105 +\tikznumberofchildren=\count106 +\tikznumberofcurrentchild=\count107 +\tikz@fig@count=\count108 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfmatrixcurrentrow=\count109 +\pgfmatrixcurrentcolumn=\count110 +\pgf@matrix@numberofcolumns=\count111 +) +\tikz@expandcount=\count112 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarytopaths.code.tex +File: tikzlibrarytopaths.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) +\sa@box=\box38 +) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2019/04/01 v2.17c AMS math features +\@mathmargin=\skip43 + +For additional information on amsmath, use the `?' option. +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 AMS text + +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks28 +\ex@=\dimen155 +)) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen156 +) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 2016/03/08 v2.02 operator names +) +\inf@bad=\count113 +LaTeX Info: Redefining \frac on input line 227. +\uproot@=\count114 +\leftroot@=\count115 +LaTeX Info: Redefining \overline on input line 389. +\classnum@=\count116 +\DOTSCASE@=\count117 +LaTeX Info: Redefining \ldots on input line 486. +LaTeX Info: Redefining \dots on input line 489. +LaTeX Info: Redefining \cdots on input line 610. +\Mathstrutbox@=\box39 +\strutbox@=\box40 +\big@size=\dimen157 +LaTeX Font Info: Redeclaring font encoding OML on input line 733. +LaTeX Font Info: Redeclaring font encoding OMS on input line 734. +\macc@depth=\count118 +\c@MaxMatrixCols=\count119 +\dotsspace@=\muskip10 +\c@parentequation=\count120 +\dspbrk@lvl=\count121 +\tag@help=\toks29 +\row@=\count122 +\column@=\count123 +\maxfields@=\count124 +\andhelp@=\toks30 +\eqnshift@=\dimen158 +\alignsep@=\dimen159 +\tagshift@=\dimen160 +\tagwidth@=\dimen161 +\totwidth@=\dimen162 +\lineht@=\dimen163 +\@envbody=\toks31 +\multlinegap=\skip44 +\multlinetaggap=\skip45 +\mathdisplay@stack=\toks32 +LaTeX Info: Redefining \[ on input line 2855. +LaTeX Info: Redefining \] on input line 2856. +) +(c:/texlive/2019/texmf-dist/tex/latex/psnfss/times.sty +Package: times 2005/04/12 PSNFSS-v9.2a (SPQR) +) +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/txfonts.sty +Package: txfonts 2008/01/22 v3.2.1 +LaTeX Font Info: Redeclaring symbol font `operators' on input line 21. +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/cmr/m/n --> OT1/txr/m/n on input line 21. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/txr/m/n on input line 21. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/txr/m/n --> OT1/txr/bx/n on input line 22. +\symitalic=\mathgroup4 +LaTeX Font Info: Overwriting symbol font `italic' in version `bold' +(Font) OT1/txr/m/it --> OT1/txr/bx/it on input line 26. +LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/cmr/bx/n --> OT1/txr/bx/n on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/txr/bx/n on input line 29. +LaTeX Font Info: Redeclaring math alphabet \mathit on input line 30. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' +(Font) OT1/cmr/m/it --> OT1/txr/m/it on input line 30. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/cmr/bx/it --> OT1/txr/m/it on input line 30. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/txr/m/it --> OT1/txr/bx/it on input line 31. +LaTeX Font Info: Redeclaring math alphabet \mathsf on input line 40. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' +(Font) OT1/cmss/m/n --> OT1/txss/m/n on input line 40. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/cmss/bx/n --> OT1/txss/m/n on input line 40. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/txss/m/n --> OT1/txss/b/n on input line 41. +LaTeX Font Info: Redeclaring math alphabet \mathtt on input line 50. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' +(Font) OT1/cmtt/m/n --> OT1/txtt/m/n on input line 50. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/cmtt/m/n --> OT1/txtt/m/n on input line 50. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/txtt/m/n --> OT1/txtt/b/n on input line 51. +LaTeX Font Info: Redeclaring symbol font `letters' on input line 58. +LaTeX Font Info: Overwriting symbol font `letters' in version `normal' +(Font) OML/cmm/m/it --> OML/txmi/m/it on input line 58. +LaTeX Font Info: Overwriting symbol font `letters' in version `bold' +(Font) OML/cmm/b/it --> OML/txmi/m/it on input line 58. +LaTeX Font Info: Overwriting symbol font `letters' in version `bold' +(Font) OML/txmi/m/it --> OML/txmi/bx/it on input line 59. +\symlettersA=\mathgroup5 +LaTeX Font Info: Overwriting symbol font `lettersA' in version `bold' +(Font) U/txmia/m/it --> U/txmia/bx/it on input line 67. +LaTeX Font Info: Redeclaring symbol font `symbols' on input line 77. +LaTeX Font Info: Overwriting symbol font `symbols' in version `normal' +(Font) OMS/cmsy/m/n --> OMS/txsy/m/n on input line 77. +LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' +(Font) OMS/cmsy/b/n --> OMS/txsy/m/n on input line 77. +LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' +(Font) OMS/txsy/m/n --> OMS/txsy/bx/n on input line 78. +\symAMSa=\mathgroup6 +LaTeX Font Info: Overwriting symbol font `AMSa' in version `bold' +(Font) U/txsya/m/n --> U/txsya/bx/n on input line 94. +\symAMSb=\mathgroup7 +LaTeX Font Info: Overwriting symbol font `AMSb' in version `bold' +(Font) U/txsyb/m/n --> U/txsyb/bx/n on input line 103. +\symsymbolsC=\mathgroup8 +LaTeX Font Info: Overwriting symbol font `symbolsC' in version `bold' +(Font) U/txsyc/m/n --> U/txsyc/bx/n on input line 113. +LaTeX Font Info: Redeclaring symbol font `largesymbols' on input line 120. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `normal' +(Font) OMX/cmex/m/n --> OMX/txex/m/n on input line 120. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' +(Font) OMX/cmex/m/n --> OMX/txex/m/n on input line 120. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' +(Font) OMX/txex/m/n --> OMX/txex/bx/n on input line 121. +\symlargesymbolsA=\mathgroup9 +LaTeX Font Info: Overwriting symbol font `largesymbolsA' in version `bold' +(Font) U/txexa/m/n --> U/txexa/bx/n on input line 129. +LaTeX Font Info: Redeclaring math symbol \mathsterling on input line 164. +LaTeX Font Info: Redeclaring math symbol \hbar on input line 591. +LaTeX Info: Redefining \not on input line 1043. +) +(c:/texlive/2019/texmf-dist/tex/latex/pgfplots/pgfplots.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.revision.tex) +Package: pgfplots 2018/03/28 v1.16 Data Visualization (1.16) + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotscore.code.tex +\t@pgfplots@toka=\toks33 +\t@pgfplots@tokb=\toks34 +\t@pgfplots@tokc=\toks35 +\pgfplots@tmpa=\dimen164 +\c@pgfplots@coordindex=\count125 +\c@pgfplots@scanlineindex=\count126 + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/sys/pgfplotssysgeneric.code.te +x)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/libs/pgfplotslibrary.code.tex) + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/oldpgfcompatib/pgfplotsoldpgfs +upp_loader.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryfpu.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsutil.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsliststru +cture.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsliststru +ctureext.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsarray.co +de.tex +\c@pgfplotsarray@tmp=\count127 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsmatrix.c +ode.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/numtable/pgfplotstableshared.c +ode.tex +\c@pgfplotstable@counta=\count128 +\t@pgfplotstable@a=\toks36 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsdeque.co +de.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsbinary.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsbinary.data.code. +tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsutil.verb.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/libs/pgflibrarypgfplots.surfsh +ading.code.tex +\c@pgfplotslibrarysurf@no=\count129 + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/sys/pgflibrarypgfplots.surfsha +ding.pgfsys-xetex.def +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/sys/pgflibrarypgfplots.surfsha +ding.pgfsys-dvipdfmx.def +\c@pgfplotslibrarysurf@streamlen=\count130 +)))) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotscolormap.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotscolor.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsstackedplots.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsplothandlers.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsmeshplothandler.code.t +ex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsmeshplotimage.code.tex +))) (c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.scaling.code.tex) + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotscoordprocessing.code.t +ex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.errorbars.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.markers.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsticks.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.paths.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarydecorations.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduledecorations.code.t +ex +\pgfdecoratedcompleteddistance=\dimen165 +\pgfdecoratedremainingdistance=\dimen166 +\pgfdecoratedinputsegmentcompleteddistance=\dimen167 +\pgfdecoratedinputsegmentremainingdistance=\dimen168 +\pgf@decorate@distancetomove=\dimen169 +\pgf@decorate@repeatstate=\count131 +\pgfdecorationsegmentamplitude=\dimen170 +\pgfdecorationsegmentlength=\dimen171 +) +\tikz@lib@dec@box=\box41 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarydecorations.pathmorphing.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/decorations/pgflibrarydec +orations.pathmorphing.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarydecorations.pathreplacing.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/decorations/pgflibrarydec +orations.pathreplacing.code.tex)) +\pgfplots@numplots=\count132 +\pgfplots@xmin@reg=\dimen172 +\pgfplots@xmax@reg=\dimen173 +\pgfplots@ymin@reg=\dimen174 +\pgfplots@ymax@reg=\dimen175 +\pgfplots@zmin@reg=\dimen176 +\pgfplots@zmax@reg=\dimen177 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryplotmarks.code.tex +File: tikzlibraryplotmarks.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplotmarks.code. +tex +File: pgflibraryplotmarks.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) (c:/texlive/2019/texmf-dist/tex/latex/csvsimple/csvsimple.sty +Package: csvsimple 2019/04/09 version 1.21 LaTeX CSV file processing + +(c:/texlive/2019/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2014/09/29 v1.1c Standard LaTeX ifthen package (DPC) +) +(c:/texlive/2019/texmf-dist/tex/latex/etoolbox/etoolbox.sty +Package: etoolbox 2019/09/21 v2.5h e-TeX tools for LaTeX (JAW) +\etb@tempcnta=\count133 +) +\csv@file=\read2 +\c@csvinputline=\count134 +\c@csvrow=\count135 +\c@csvcol=\count136 +\csv@out=\write4 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryarrows.code.tex +File: tikzlibraryarrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex +File: pgflibraryarrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\arrowsize=\dimen178 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryintersections.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryintersections.c +ode.tex +\pgf@intersect@solutions=\count137 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarymath.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryfpu.code.tex) +\tikz@math@for@depth=\count138 +\tikz@math@dimen=\dimen179 +\tikz@math@toks=\toks37 +) (./polynom1.aux) +\openout1 = `polynom1.aux'. + +LaTeX Font Info: Checking defaults for OML/txmi/m/it on input line 11. +LaTeX Font Info: Trying to load font information for OML+txmi on input line +11. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/omltxmi.fd +File: omltxmi.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 11. +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 11. +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for OMS/txsy/m/n on input line 11. +LaTeX Font Info: Trying to load font information for OMS+txsy on input line +11. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/omstxsy.fd +File: omstxsy.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 11. +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for OMX/txex/m/n on input line 11. +LaTeX Font Info: Trying to load font information for OMX+txex on input line +11. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/omxtxex.fd +File: omxtxex.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for U/txexa/m/n on input line 11. +LaTeX Font Info: Trying to load font information for U+txexa on input line 1 +1. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxexa.fd +File: utxexa.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Trying to load font information for TU+txr on input line 11 +. +LaTeX Font Info: No file TUtxr.fd. on input line 11. + + +LaTeX Font Warning: Font shape `TU/txr/m/n' undefined +(Font) using `TU/lmr/m/n' instead on input line 11. + +ABD: EveryShipout initializing macros + +Package pgfplots Warning: running in backwards compatibility mode (unsuitable t +ick labels; missing features). Consider writing \pgfplotsset{compat=1.16} into +your preamble. + on input line 11. + +LaTeX Font Info: Trying to load font information for OT1+txr on input line 2 +6. +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/ot1txr.fd +File: ot1txr.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txmia on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxmia.fd +File: utxmia.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txsya on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxsya.fd +File: utxsya.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txsyb on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxsyb.fd +File: utxsyb.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txsyc on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxsyc.fd +File: utxsyc.fd 2000/12/15 v3.1 +) [1 + +] (./polynom1.aux) + +LaTeX Font Warning: Some font shapes were not available, defaults substituted. + + ) +Here is how much of TeX's memory you used: + 25076 strings out of 492483 + 634297 string characters out of 6132858 + 794987 words of memory out of 5000000 + 29148 multiletter control sequences out of 15000+600000 + 13575 words of font info for 45 fonts, out of 8000000 for 9000 + 1348 hyphenation exceptions out of 8191 + 68i,6n,98p,840b,724s stack positions out of 5000i,500n,10000p,200000b,80000s + +Output written on polynom1.pdf (1 page). diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.pdf b/buch/papers/reedsolomon/RS presentation/images/polynom1.pdf Binary files differnew file mode 100644 index 0000000..1f632d8 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz b/buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz Binary files differnew file mode 100644 index 0000000..578f2a3 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.tex b/buch/papers/reedsolomon/RS presentation/images/polynom1.tex new file mode 100644 index 0000000..db83daa --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.tex @@ -0,0 +1,59 @@ +% polynome1 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\teiler}{40} +\begin{document} + %Übertragen von den Zahlen + %\textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} + %als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline + %Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + % \textcolor{green}{15}, \textcolor{green}{26}, + % \textcolor{green}{ 41}, \textcolor{green}{60}, + % \textcolor{green}{83}, \textcolor{green}{110})$ + + +\begin{tikzpicture}[>=latex,thick] + +\draw[color=blue, line width=1.4pt] +plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); +\draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; +\draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; +\def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; +} +\punkt{(1,8/\teiler)} +\punkt{(2,15/\teiler)} +\punkt{(3,26/\teiler)} +\punkt{(4,41/\teiler)} +\punkt{(5,60/\teiler)} +\punkt{(6,83/\teiler)} +\punkt{(7,110/\teiler)} +%\draw[color=gray,line width=1pt,dashed] +%plot[domain=0.5:7, samples=100] +%({\x},{(0.1958*\x^2-1.2875*\x+3.0417)}); +%\def\erpunkt#1{ +% \fill[color=red] #1 circle[radius=0.08]; +% \draw #1 circle[radius=0.07]; +%} +%\erpunkt{(2,50/\teiler)} +%\erpunkt{(3,0.9414)} +%\punkt{(4,41/\teiler)} +%\punkt{(5,60/\teiler)} + +\draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; +\draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; + + + + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.aux b/buch/papers/reedsolomon/RS presentation/images/polynom2.aux new file mode 100644 index 0000000..f23e546 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.aux @@ -0,0 +1 @@ +\relax diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.log b/buch/papers/reedsolomon/RS presentation/images/polynom2.log new file mode 100644 index 0000000..169203e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.log @@ -0,0 +1,747 @@ +This is XeTeX, Version 3.14159265-2.6-0.999991 (TeX Live 2019/W32TeX) (preloaded format=xelatex 2019.10.25) 21 APR 2021 12:28 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**polynom2.tex +(./polynom2.tex +LaTeX2e <2019-10-01> patch level 1 +(c:/texlive/2019/texmf-dist/tex/latex/standalone/standalone.cls +Document Class: standalone 2018/03/26 v1.3a Class to compile TeX sub-files stan +dalone +(c:/texlive/2019/texmf-dist/tex/latex/tools/shellesc.sty +Package: shellesc 2016/06/07 v0.02b unified shell escape interface for LaTeX +Package shellesc Info: Restricted shell escape enabled on input line 72. +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifluatex.sty +Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch +) +(c:/texlive/2019/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +(c:/texlive/2019/texmf-dist/tex/latex/xkeyval/xkeyval.sty +Package: xkeyval 2014/12/03 v2.7a package option processing (HA) + +(c:/texlive/2019/texmf-dist/tex/generic/xkeyval/xkeyval.tex +(c:/texlive/2019/texmf-dist/tex/generic/xkeyval/xkvutils.tex +\XKV@toks=\toks14 +\XKV@tempa@toks=\toks15 + +(c:/texlive/2019/texmf-dist/tex/generic/xkeyval/keyval.tex)) +\XKV@depth=\count80 +File: xkeyval.tex 2014/12/03 v2.7a key=value parser (HA) +)) +\sa@internal=\count81 +\c@sapage=\count82 + +(c:/texlive/2019/texmf-dist/tex/latex/standalone/standalone.cfg +File: standalone.cfg 2018/03/26 v1.3a Default configuration file for 'standalon +e' class +) +(c:/texlive/2019/texmf-dist/tex/latex/base/article.cls +Document Class: article 2019/08/27 v1.4j Standard LaTeX document class +(c:/texlive/2019/texmf-dist/tex/latex/base/size10.clo +File: size10.clo 2019/08/27 v1.4j Standard LaTeX file (size option) +) +\c@part=\count83 +\c@section=\count84 +\c@subsection=\count85 +\c@subsubsection=\count86 +\c@paragraph=\count87 +\c@subparagraph=\count88 +\c@figure=\count89 +\c@table=\count90 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +\pgfutil@everybye=\toks16 +\pgfutil@tempdima=\dimen103 +\pgfutil@tempdimb=\dimen104 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box27 + +(c:/texlive/2019/texmf-dist/tex/latex/ms/everyshi.sty +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/pgf.revision.tex) +Package: pgfrcs 2019/08/03 v3.1.4b (3.1.4b) +)) +Package: pgf 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2019/10/08 v1.3c Standard LaTeX Graphics (DPC,SPQR) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 2016/01/03 v1.10 sin cos tan (DPC) +) +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration +) +Package graphics Info: Driver file: xetex.def on input line 105. + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-def/xetex.def +File: xetex.def 2017/06/24 v5.0h Graphics/color driver for xetex +)) +\Gin@req@height=\dimen105 +\Gin@req@width=\dimen106 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks17 +\pgfkeys@temptoks=\toks18 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +\pgfkeys@tmptoks=\toks19 +)) +\pgf@x=\dimen107 +\pgf@y=\dimen108 +\pgf@xa=\dimen109 +\pgf@ya=\dimen110 +\pgf@xb=\dimen111 +\pgf@yb=\dimen112 +\pgf@xc=\dimen113 +\pgf@yc=\dimen114 +\pgf@xd=\dimen115 +\pgf@yd=\dimen116 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count91 +\c@pgf@countb=\count92 +\c@pgf@countc=\count93 +\c@pgf@countd=\count94 +\t@pgf@toka=\toks20 +\t@pgf@tokb=\toks21 +\t@pgf@tokc=\toks22 +\pgf@sys@id@count=\count95 + (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2019/08/03 v3.1.4b (3.1.4b) +) +Driver file for pgf: pgfsys-xetex.def + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-xetex.def +File: pgfsys-xetex.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-dvipdfmx.def +File: pgfsys-dvipdfmx.def 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +File: pgfsys-common-pdf.def 2019/08/03 v3.1.4b (3.1.4b) +) +\pgfsys@objnum=\count96 +))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +File: pgfsyssoftpath.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfsyssoftpath@smallbuffer@items=\count97 +\pgfsyssoftpath@bigbuffer@items=\count98 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +File: pgfsysprotocol.code.tex 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/xcolor/xcolor.sty +Package: xcolor 2016/05/11 v2.12 LaTeX color extensions (UK) + +(c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/color.cfg +File: color.cfg 2016/01/02 v1.6 sample color configuration +) +Package xcolor Info: Driver file: xetex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1348. +Package xcolor Info: Model `RGB' extended on input line 1364. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1366. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1367. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1368. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1369. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1370. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1371. +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen117 +\pgfmath@count=\count99 +\pgfmath@box=\box28 +\pgfmath@toks=\toks23 +\pgfmath@stack@operand=\toks24 +\pgfmath@stack@operation=\toks25 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric +.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.t +ex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.co +de.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithm +etics.code.tex))) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count100 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +File: pgfcorepoints.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@picminx=\dimen118 +\pgf@picmaxx=\dimen119 +\pgf@picminy=\dimen120 +\pgf@picmaxy=\dimen121 +\pgf@pathminx=\dimen122 +\pgf@pathmaxx=\dimen123 +\pgf@pathminy=\dimen124 +\pgf@pathmaxy=\dimen125 +\pgf@xx=\dimen126 +\pgf@xy=\dimen127 +\pgf@yx=\dimen128 +\pgf@yy=\dimen129 +\pgf@zx=\dimen130 +\pgf@zy=\dimen131 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.cod +e.tex +File: pgfcorepathconstruct.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@path@lastx=\dimen132 +\pgf@path@lasty=\dimen133 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.te +x +File: pgfcorepathusage.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@shorten@end@additional=\dimen134 +\pgf@shorten@start@additional=\dimen135 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +File: pgfcorescopes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfpic=\box29 +\pgf@hbox=\box30 +\pgf@layerbox@main=\box31 +\pgf@picture@serial@count=\count101 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code +.tex +File: pgfcoregraphicstate.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgflinewidth=\dimen136 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.c +ode.tex +File: pgfcoretransformations.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@pt@x=\dimen137 +\pgf@pt@y=\dimen138 +\pgf@pt@temp=\dimen139 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +File: pgfcorequick.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +File: pgfcoreobjects.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.co +de.tex +File: pgfcorepathprocessing.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +File: pgfcorearrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfarrowsep=\dimen140 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +File: pgfcoreshade.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@max=\dimen141 +\pgf@sys@shading@range@num=\count102 +\pgf@shadingcount=\count103 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +File: pgfcoreimage.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +File: pgfcoreexternal.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfexternal@startupbox=\box32 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +File: pgfcorelayers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code +.tex +File: pgfcoretransparency.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +File: pgfcorepatterns.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +File: pgfcorerdf.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) (c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.te +x +File: pgfmoduleshapes.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfnodeparttextbox=\box33 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +File: pgfmoduleplot.code.tex 2019/08/03 v3.1.4b (3.1.4b) +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.st +y +Package: pgfcomp-version-0-65 2019/08/03 v3.1.4b (3.1.4b) +\pgf@nodesepstart=\dimen142 +\pgf@nodesepend=\dimen143 +) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.st +y +Package: pgfcomp-version-1-18 2019/08/03 v3.1.4b (3.1.4b) +)) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +(c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) +(c:/texlive/2019/texmf-dist/tex/latex/pgf/math/pgfmath.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex) +\pgffor@iter=\dimen144 +\pgffor@skip=\dimen145 +\pgffor@stack=\toks26 +\pgffor@toks=\toks27 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +Package: tikz 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.co +de.tex +File: pgflibraryplothandlers.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgf@plot@mark@count=\count104 +\pgfplotmarksize=\dimen146 +) +\tikz@lastx=\dimen147 +\tikz@lasty=\dimen148 +\tikz@lastxsaved=\dimen149 +\tikz@lastysaved=\dimen150 +\tikz@lastmovetox=\dimen151 +\tikz@lastmovetoy=\dimen152 +\tikzleveldistance=\dimen153 +\tikzsiblingdistance=\dimen154 +\tikz@figbox=\box34 +\tikz@figbox@bg=\box35 +\tikz@tempbox=\box36 +\tikz@tempbox@bg=\box37 +\tikztreelevel=\count105 +\tikznumberofchildren=\count106 +\tikznumberofcurrentchild=\count107 +\tikz@fig@count=\count108 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +File: pgfmodulematrix.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\pgfmatrixcurrentrow=\count109 +\pgfmatrixcurrentcolumn=\count110 +\pgf@matrix@numberofcolumns=\count111 +) +\tikz@expandcount=\count112 + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarytopaths.code.tex +File: tikzlibrarytopaths.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) +\sa@box=\box38 +) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2019/04/01 v2.17c AMS math features +\@mathmargin=\skip43 + +For additional information on amsmath, use the `?' option. +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 AMS text + +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks28 +\ex@=\dimen155 +)) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen156 +) +(c:/texlive/2019/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 2016/03/08 v2.02 operator names +) +\inf@bad=\count113 +LaTeX Info: Redefining \frac on input line 227. +\uproot@=\count114 +\leftroot@=\count115 +LaTeX Info: Redefining \overline on input line 389. +\classnum@=\count116 +\DOTSCASE@=\count117 +LaTeX Info: Redefining \ldots on input line 486. +LaTeX Info: Redefining \dots on input line 489. +LaTeX Info: Redefining \cdots on input line 610. +\Mathstrutbox@=\box39 +\strutbox@=\box40 +\big@size=\dimen157 +LaTeX Font Info: Redeclaring font encoding OML on input line 733. +LaTeX Font Info: Redeclaring font encoding OMS on input line 734. +\macc@depth=\count118 +\c@MaxMatrixCols=\count119 +\dotsspace@=\muskip10 +\c@parentequation=\count120 +\dspbrk@lvl=\count121 +\tag@help=\toks29 +\row@=\count122 +\column@=\count123 +\maxfields@=\count124 +\andhelp@=\toks30 +\eqnshift@=\dimen158 +\alignsep@=\dimen159 +\tagshift@=\dimen160 +\tagwidth@=\dimen161 +\totwidth@=\dimen162 +\lineht@=\dimen163 +\@envbody=\toks31 +\multlinegap=\skip44 +\multlinetaggap=\skip45 +\mathdisplay@stack=\toks32 +LaTeX Info: Redefining \[ on input line 2855. +LaTeX Info: Redefining \] on input line 2856. +) +(c:/texlive/2019/texmf-dist/tex/latex/psnfss/times.sty +Package: times 2005/04/12 PSNFSS-v9.2a (SPQR) +) +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/txfonts.sty +Package: txfonts 2008/01/22 v3.2.1 +LaTeX Font Info: Redeclaring symbol font `operators' on input line 21. +LaTeX Font Info: Overwriting symbol font `operators' in version `normal' +(Font) OT1/cmr/m/n --> OT1/txr/m/n on input line 21. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/txr/m/n on input line 21. +LaTeX Font Info: Overwriting symbol font `operators' in version `bold' +(Font) OT1/txr/m/n --> OT1/txr/bx/n on input line 22. +\symitalic=\mathgroup4 +LaTeX Font Info: Overwriting symbol font `italic' in version `bold' +(Font) OT1/txr/m/it --> OT1/txr/bx/it on input line 26. +LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/cmr/bx/n --> OT1/txr/bx/n on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/txr/bx/n on input line 29. +LaTeX Font Info: Redeclaring math alphabet \mathit on input line 30. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' +(Font) OT1/cmr/m/it --> OT1/txr/m/it on input line 30. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/cmr/bx/it --> OT1/txr/m/it on input line 30. +LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' +(Font) OT1/txr/m/it --> OT1/txr/bx/it on input line 31. +LaTeX Font Info: Redeclaring math alphabet \mathsf on input line 40. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' +(Font) OT1/cmss/m/n --> OT1/txss/m/n on input line 40. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/cmss/bx/n --> OT1/txss/m/n on input line 40. +LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' +(Font) OT1/txss/m/n --> OT1/txss/b/n on input line 41. +LaTeX Font Info: Redeclaring math alphabet \mathtt on input line 50. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' +(Font) OT1/cmtt/m/n --> OT1/txtt/m/n on input line 50. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/cmtt/m/n --> OT1/txtt/m/n on input line 50. +LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' +(Font) OT1/txtt/m/n --> OT1/txtt/b/n on input line 51. +LaTeX Font Info: Redeclaring symbol font `letters' on input line 58. +LaTeX Font Info: Overwriting symbol font `letters' in version `normal' +(Font) OML/cmm/m/it --> OML/txmi/m/it on input line 58. +LaTeX Font Info: Overwriting symbol font `letters' in version `bold' +(Font) OML/cmm/b/it --> OML/txmi/m/it on input line 58. +LaTeX Font Info: Overwriting symbol font `letters' in version `bold' +(Font) OML/txmi/m/it --> OML/txmi/bx/it on input line 59. +\symlettersA=\mathgroup5 +LaTeX Font Info: Overwriting symbol font `lettersA' in version `bold' +(Font) U/txmia/m/it --> U/txmia/bx/it on input line 67. +LaTeX Font Info: Redeclaring symbol font `symbols' on input line 77. +LaTeX Font Info: Overwriting symbol font `symbols' in version `normal' +(Font) OMS/cmsy/m/n --> OMS/txsy/m/n on input line 77. +LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' +(Font) OMS/cmsy/b/n --> OMS/txsy/m/n on input line 77. +LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' +(Font) OMS/txsy/m/n --> OMS/txsy/bx/n on input line 78. +\symAMSa=\mathgroup6 +LaTeX Font Info: Overwriting symbol font `AMSa' in version `bold' +(Font) U/txsya/m/n --> U/txsya/bx/n on input line 94. +\symAMSb=\mathgroup7 +LaTeX Font Info: Overwriting symbol font `AMSb' in version `bold' +(Font) U/txsyb/m/n --> U/txsyb/bx/n on input line 103. +\symsymbolsC=\mathgroup8 +LaTeX Font Info: Overwriting symbol font `symbolsC' in version `bold' +(Font) U/txsyc/m/n --> U/txsyc/bx/n on input line 113. +LaTeX Font Info: Redeclaring symbol font `largesymbols' on input line 120. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `normal' +(Font) OMX/cmex/m/n --> OMX/txex/m/n on input line 120. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' +(Font) OMX/cmex/m/n --> OMX/txex/m/n on input line 120. +LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' +(Font) OMX/txex/m/n --> OMX/txex/bx/n on input line 121. +\symlargesymbolsA=\mathgroup9 +LaTeX Font Info: Overwriting symbol font `largesymbolsA' in version `bold' +(Font) U/txexa/m/n --> U/txexa/bx/n on input line 129. +LaTeX Font Info: Redeclaring math symbol \mathsterling on input line 164. +LaTeX Font Info: Redeclaring math symbol \hbar on input line 591. +LaTeX Info: Redefining \not on input line 1043. +) +(c:/texlive/2019/texmf-dist/tex/latex/pgfplots/pgfplots.sty +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.revision.tex) +Package: pgfplots 2018/03/28 v1.16 Data Visualization (1.16) + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotscore.code.tex +\t@pgfplots@toka=\toks33 +\t@pgfplots@tokb=\toks34 +\t@pgfplots@tokc=\toks35 +\pgfplots@tmpa=\dimen164 +\c@pgfplots@coordindex=\count125 +\c@pgfplots@scanlineindex=\count126 + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/sys/pgfplotssysgeneric.code.te +x)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/libs/pgfplotslibrary.code.tex) + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/oldpgfcompatib/pgfplotsoldpgfs +upp_loader.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryfpu.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsutil.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsliststru +cture.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsliststru +ctureext.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsarray.co +de.tex +\c@pgfplotsarray@tmp=\count127 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsmatrix.c +ode.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/numtable/pgfplotstableshared.c +ode.tex +\c@pgfplotstable@counta=\count128 +\t@pgfplotstable@a=\toks36 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/liststructure/pgfplotsdeque.co +de.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsbinary.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsbinary.data.code. +tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotsutil.verb.code.te +x) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/libs/pgflibrarypgfplots.surfsh +ading.code.tex +\c@pgfplotslibrarysurf@no=\count129 + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/sys/pgflibrarypgfplots.surfsha +ding.pgfsys-xetex.def +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/sys/pgflibrarypgfplots.surfsha +ding.pgfsys-dvipdfmx.def +\c@pgfplotslibrarysurf@streamlen=\count130 +)))) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotscolormap.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/util/pgfplotscolor.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsstackedplots.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsplothandlers.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsmeshplothandler.code.t +ex +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsmeshplotimage.code.tex +))) (c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.scaling.code.tex) + +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotscoordprocessing.code.t +ex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.errorbars.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.markers.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplotsticks.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgfplots/pgfplots.paths.code.tex) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarydecorations.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmoduledecorations.code.t +ex +\pgfdecoratedcompleteddistance=\dimen165 +\pgfdecoratedremainingdistance=\dimen166 +\pgfdecoratedinputsegmentcompleteddistance=\dimen167 +\pgfdecoratedinputsegmentremainingdistance=\dimen168 +\pgf@decorate@distancetomove=\dimen169 +\pgf@decorate@repeatstate=\count131 +\pgfdecorationsegmentamplitude=\dimen170 +\pgfdecorationsegmentlength=\dimen171 +) +\tikz@lib@dec@box=\box41 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarydecorations.pathmorphing.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/decorations/pgflibrarydec +orations.pathmorphing.code.tex)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarydecorations.pathreplacing.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/decorations/pgflibrarydec +orations.pathreplacing.code.tex)) +\pgfplots@numplots=\count132 +\pgfplots@xmin@reg=\dimen172 +\pgfplots@xmax@reg=\dimen173 +\pgfplots@ymin@reg=\dimen174 +\pgfplots@ymax@reg=\dimen175 +\pgfplots@zmin@reg=\dimen176 +\pgfplots@zmax@reg=\dimen177 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryplotmarks.code.tex +File: tikzlibraryplotmarks.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplotmarks.code. +tex +File: pgflibraryplotmarks.code.tex 2019/08/03 v3.1.4b (3.1.4b) +))) (c:/texlive/2019/texmf-dist/tex/latex/csvsimple/csvsimple.sty +Package: csvsimple 2019/04/09 version 1.21 LaTeX CSV file processing + +(c:/texlive/2019/texmf-dist/tex/latex/base/ifthen.sty +Package: ifthen 2014/09/29 v1.1c Standard LaTeX ifthen package (DPC) +) +(c:/texlive/2019/texmf-dist/tex/latex/etoolbox/etoolbox.sty +Package: etoolbox 2019/09/21 v2.5h e-TeX tools for LaTeX (JAW) +\etb@tempcnta=\count133 +) +\csv@file=\read2 +\c@csvinputline=\count134 +\c@csvrow=\count135 +\c@csvcol=\count136 +\csv@out=\write4 +) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryarrows.code.tex +File: tikzlibraryarrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex +File: pgflibraryarrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) +\arrowsize=\dimen178 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryintersections.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryintersections.c +ode.tex +\pgf@intersect@solutions=\count137 +)) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +brarymath.code.tex +(c:/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzli +braryfpu.code.tex) +\tikz@math@for@depth=\count138 +\tikz@math@dimen=\dimen179 +\tikz@math@toks=\toks37 +) (./polynom2.aux) +\openout1 = `polynom2.aux'. + +LaTeX Font Info: Checking defaults for OML/txmi/m/it on input line 11. +LaTeX Font Info: Trying to load font information for OML+txmi on input line +11. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/omltxmi.fd +File: omltxmi.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 11. +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 11. +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for OMS/txsy/m/n on input line 11. +LaTeX Font Info: Trying to load font information for OMS+txsy on input line +11. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/omstxsy.fd +File: omstxsy.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 11. +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for OMX/txex/m/n on input line 11. +LaTeX Font Info: Trying to load font information for OMX+txex on input line +11. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/omxtxex.fd +File: omxtxex.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Checking defaults for U/txexa/m/n on input line 11. +LaTeX Font Info: Trying to load font information for U+txexa on input line 1 +1. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxexa.fd +File: utxexa.fd 2000/12/15 v3.1 +) +LaTeX Font Info: ... okay on input line 11. +LaTeX Font Info: Trying to load font information for TU+txr on input line 11 +. +LaTeX Font Info: No file TUtxr.fd. on input line 11. + + +LaTeX Font Warning: Font shape `TU/txr/m/n' undefined +(Font) using `TU/lmr/m/n' instead on input line 11. + +ABD: EveryShipout initializing macros + +Package pgfplots Warning: running in backwards compatibility mode (unsuitable t +ick labels; missing features). Consider writing \pgfplotsset{compat=1.16} into +your preamble. + on input line 11. + +LaTeX Font Info: Trying to load font information for OT1+txr on input line 2 +6. +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/ot1txr.fd +File: ot1txr.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txmia on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxmia.fd +File: utxmia.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txsya on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxsya.fd +File: utxsya.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txsyb on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxsyb.fd +File: utxsyb.fd 2000/12/15 v3.1 +) +LaTeX Font Info: Trying to load font information for U+txsyc on input line 2 +6. + +(c:/texlive/2019/texmf-dist/tex/latex/txfonts/utxsyc.fd +File: utxsyc.fd 2000/12/15 v3.1 +) [1 + +] (./polynom2.aux) + +LaTeX Font Warning: Some font shapes were not available, defaults substituted. + + ) +Here is how much of TeX's memory you used: + 25292 strings out of 492483 + 638772 string characters out of 6132858 + 794987 words of memory out of 5000000 + 29364 multiletter control sequences out of 15000+600000 + 13575 words of font info for 45 fonts, out of 8000000 for 9000 + 1348 hyphenation exceptions out of 8191 + 68i,6n,98p,840b,724s stack positions out of 5000i,500n,10000p,200000b,80000s + +Output written on polynom2.pdf (1 page). diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf b/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf Binary files differnew file mode 100644 index 0000000..05f4ba0 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz b/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz Binary files differnew file mode 100644 index 0000000..24859aa --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.tex b/buch/papers/reedsolomon/RS presentation/images/polynom2.tex new file mode 100644 index 0000000..aa792ce --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.tex @@ -0,0 +1,57 @@ +% polynome2 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\teiler}{40} +\begin{document} + %Übertragen von den Zahlen + %\textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} + %als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline + %Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + % \textcolor{green}{15}, \textcolor{green}{26}, + % \textcolor{green}{ 41}, \textcolor{green}{60}, + % \textcolor{green}{83}, \textcolor{green}{110})$ + + + \begin{tikzpicture}[>=latex,thick] + + \draw[color=blue, line width=1.4pt] + plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); + \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; + \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; + \def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \punkt{(1,8/\teiler)} + %\punkt{(2,15/\teiler)} + %\punkt{(3,26/\teiler)} + \punkt{(4,41/\teiler)} + \punkt{(5,60/\teiler)} + \punkt{(6,83/\teiler)} + \punkt{(7,110/\teiler)} + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(0.1958*\x^2-1.2875*\x+3.0417)}); + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,0.9414)} + + + \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; + \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; + + + + + \end{tikzpicture} +\end{document} diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex new file mode 100644 index 0000000..262297e --- /dev/null +++ b/buch/papers/reedsolomon/codebsp.tex @@ -0,0 +1,181 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Codierung eines Beispiels +\label{reedsolomon:section:codebsp}} +\rhead{Koerper Festlegen} + +Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir die einzelnen Probleme und ihre Lösungen anhand eines Beispiels betrachten. +Da wir in Endlichen Körpern Rechnen werden wir zuerst solch einen Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6 (wie verweist man auf eine definition?)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. +Wir legen für unser Beispiel den endlichen Körper mit $q = 11$ fest. +Zur Hilfestellung können dazu die beiden Tabellen \ref{reedsolomon:subsection:adtab} und +\ref{reedsolomon:subsection:mptab} hinzugezogen werden. Diese Tabellen enthalten sämtliche Resultate aller gültigen Operationen \textcolor{red}{(Notiz: nach meinem Wissen gibt es ja nur addition und multiplikation als gültige operationen)}, die in diesem Körper durchgeführt werden können. +Aus der Definition der Endlichen Körper (ersichtlich auch in den Tabellen) folgt, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur verfügung stehen und somit $11 = 0$ gelten muss. + +% OLD TEXT +%Alle folgenden Berechnungen wurden mit den beiden Restetabellen \ref{reedsolomon:subsection:adtab} und \ref{reedsolomon:subsection:mptab} durchgeführt. +%Aus den Tabellen folgt auch, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen. + +% die beiden Restetabellen von F_11 +%\input{papers/reedsolomon/restetabelle1} +%\input{papers/reedsolomon/restetabelle2} + +Anhand der Menge uns zur Verfügung stehenden Zahlen wird auch festgelegt, wie viele Zahlen ein Nachrichtenblock $n$, bestehend aus Nutzdatenteil und Fehlerkorrekturteil, umfassen kann. +Der Nachrichtenblock im Beispiel besteht aus +\[ +n = q - 1 = 10 \text{ Zahlen}, +\] +wobei die null weggelassen wird. Wenn wir versuchen würden, mit der null zu codieren, so stellen wir fest, dass wir wieder null an der gleichen Stelle erhalten und somit wäre die Codierung nicht eindeutig. + +% Notes +%Da bei allen Codes, die codiert werden wird an der gleichen Stelle eine Nullstelle auftreten. + +% Old Text +%Die grösse des endlichen Körpers legt auch fest, wie gross unsere Nachricht $n$ bestehend aus Nutzdatenteil und Fehlerkorrekturteil sein kann und beträgt in unserem Beispiel +%\[ +%n = q - 1 = 10 \text{ Zahlen}. +%\] + +Im nächsten Schritt bestimmen wir, wie viele Fehler $t$ maximal während der Übertragung auftreten dürfen, damit wir sie noch korrigieren können. +Unser Beispielcode sollte in der Lage sein +\[ +t = 2 +\] +Fehlerstellen korrigieren zu können. + +Die Grösse des Nutzdatenteils hängt von der Grösse des Nachrichtenblocks sowie der Anzahl der Fehlerkorrekturstellen ab. Je robuster der Code sein muss, desto weniger Platz für Nutzdaten $k$ bleibt in der Nachricht übrig. +Bei maximal 2 Fehler können wir noch +\[ +k = n - 2t = 6\text{ Zahlen} +\] +übertragen. + +Zusammenfassend haben wir einen Nachrichtenblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten zu rekonstruieren. Zudem werden wir im weiteren feststellen, dass dieser Code maximal vier Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. + +Wir legen nun die Nachricht +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Stellen für die Fehlerkorrektur zuständig sind. +Solange diese Stellen vor dem Codieren und nach dem Decodieren den Wert null haben, so ist die Nachricht Fehlerfrei übertragen worden. + +Da wir in den folgenden Abschnitten mit Polynomen arbeiten, stellen wir die Nachicht auch noch als Polynom +\[ +m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 +\] +dar. + +% Old Text +%Die Nachricht können wir auch als Polynom +%\[ +%m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 +%\] +%darstellen. + +\subsection{Der Ansatz der diskreten Fouriertransformation + \label{reedsolomon:subsection:diskFT}} + +In einem vorherigen Kapitel \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. +Wir legen deshalb die Zahl $a$ fest. Diese Zahl soll die gleichen aufgaben haben, wie $e^{\frac{j}{2 \pi}}$ in der Diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ existiert. Dazu soll $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken, um +\[ +\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} +\] +in +\[ +\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. +\] +umzuschreiben. +% Old Text +%Wir suchen also eine Zahl $a$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann. +%Dazu schreiben wir +%\[ +%\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} +%\] +%um in +%\[ +%\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. +%\] +% +%Wenn wir alle möglichen Werte für $a$ einsetzen, also +%\begin{align} +%a = 0 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\} \\ +%a = 1 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\} \\ +%a = 2 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\} \\ +%a = 3 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\} \\ +%a = 4 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\} \\ +%a = 5 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\} \\ +%a = 6 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\} \\ +%a = 7 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\} \\ +%a = 8 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\} \\ +%a = 9 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\} \\ +%a = 10 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\} +%\end{align} + +\subsubsection{Die primitiven Einheitswurzeln + \label{reedsolomon:subsection:primsqrt}} + +Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen +\begin{center} +\begin{tabular}{c r c l} +%$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\ +$a = 1 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ \\ +$a = 2 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ \\ +$a = 3 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ \\ +$a = 4 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ \\ +$a = 5 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ \\ +$a = 6 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ \\ +$a = 7 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ \\ +$a = 8 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ \\ +$a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ \\ +$a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ +\end{tabular} +\end{center} +so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. +Wenden wir diese Vorgehensweise auch für andere Endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a^i = 8$. + +\subsubsection{Bildung einer Transformationsmatrix + \label{reedsolomon:subsection:transMat}} + +Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu Codieren. Daraus sollen wir dann einen Übertragungsvektor $v$ erhalten, den wir an den Empfänger schicken können. Für die Codierung müssen wir alle $a^i$ in das Polynom $m(X)$ einsetzen. Da wir $a^i = 8^i$ gewählt haben ergibt sich daraus +% +%Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. +% +\begin{center} + \begin{tabular}{c} + $m(8^0) = 4 \cdot 1^5 + 7 \cdot 1^4 + 2 \cdot 1^3 + 5 \cdot 1^2 + 8 \cdot 1^1 + 1 = 5$ \\ + $m(8^1) = 4 \cdot 8^5 + 7 \cdot 8^4 + 2 \cdot 8^3 + 5 \cdot 8^2 + 8 \cdot 8^1 + 1 = 3$ \\ + \vdots \\ + $m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$ + \end{tabular} +\end{center} +unser Übertragungsvektor. Um das ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen und bildet so unsere Transformationsmatrix $A$. + +\subsection{Allgemeine Codierung + \label{reedsolomon:subsection:algCod}} + +Für die Codierung benötigen wir die Nachricht $m$, die Codiert werden soll sowie die Transformationsmatrix $A$. Daraus erhalten wir den Übertragungsvektor $v$. Setzen wir die Zahlen aus dem Beispiel ein erhalten wir folgende Darstellung. +\[ +v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ +\end{pmatrix} +\] +Für unseren Übertragungsvektor resultiert +\[ +v = [5,3,6,5,2,10,2,7,10,4], +\] +den wir jetzt über einen beliebigen Nachrichtenkanal versenden können. diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex new file mode 100644 index 0000000..feaa027 --- /dev/null +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -0,0 +1,312 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Decodierung: Ansatz mit Fehlerkorrektur +\label{reedsolomon:section:decmitfehler}} +\rhead{fehlerhafte rekonstruktion} +Bisher haben wir die Decodierung unter der Bedingung durchgeführt, dass der Übertragungsvektor fehlerlos versendet und empfangen wurde. +In der realen Welt müssen wir uns jedoch damit abfinden, dass kein Übertragungskanal garantiert fehlerfrei ist und das wir früher oder später mit Fehlern rechnen müssen. +Genau für dieses Problem wurden Fehler korrigierende Codes, wie der Reed-Solomon-Code, entwickelt. +In diesem Abschnitt betrachten wir somit die Idee der Fehlerkorrektur und wie wir diese auf unser Beispiel anwenden können. +Der Übertragungskanal im Beispiel weisst jetzt den Fehlervektor +\[ +u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] +\] +auf. + +Senden wir jetzt unser Übertragungsvektor $v$ durch diesen Kanal addiert sich der Fehlervektor $u$ auf unsere Übertragung und wir erhalten +\begin{center} + + \begin{tabular}{c | c r } + $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ + $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ + \hline + $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ + \end{tabular} + + % alternative design + %\begin{tabular}{c | c cccccccccccc } + % $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ + % $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ + % \hline + % $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ + %\end{tabular} + +\end{center} +als neuen, fehlerbehafteten Übertragungsvektor $w$ auf der Empfängerseite. +% Old Text +%In diesem Abschnitt gehen wir genauer darauf ein, wie der Reed-Solomon-Code eine solche Feherkorrektur vornimt. +% +%In diesem Abschnitt betrachten wir das Problem, dass während der Übertragung des Übertragungsvektors von unserem Beispiel +% +% +%Zu diesem Zweck wurden Fehler korrigierende Codes entwickelt. +% +%Dieser Optimalfall kann jedoch mit keinem Übertragungskanal garantiert werden +% +% +%Im zweiten Teil zur Decodierung betrachten wir den Fall, dass unser Übertragungskanal nicht fehlerfrei ist. +%Wir legen daher den Fehlervektor +%\[ +%u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] +%\] +%fest, den wir zu unserem Übertragungsvektor als Fehler dazu addieren und somit +% +%\begin{center} +% +%\begin{tabular}{c | c r } +% $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ +% $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ +% \hline +% $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ +%\end{tabular} +% +%% alternative design +%%\begin{tabular}{c | c cccccccccccc } +%% $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ +%% $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ +%% \hline +%% $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ +%%\end{tabular} +% +%\end{center} +%als Übertragungsvektor auf der Empfängerseite erhalten. +Wir jetzt als Empfänger wissen jedoch nicht, dass der erhaltene Übertragungsvektor jetzt fehlerbehaftet ist und werden dementsprechend den Ansatz aus Abschnitt \ref{reedsolomon:section:decohnefehler} anwenden. +Wir stellen jedoch recht schnell fest, dass am decodierten Nachrichtenblock +\[ +r = [\underbrace{5,7,4,10,}_{\text{Syndrom}}5,4,5,7,6,7]. +\] +etwas nicht in Ordnung ist, denn die vorderen vier Fehlerkorrekturstellen haben nicht mehr den Wert null. +Der Nachrichtenblock weisst jetzt ein \em Syndrom \em auf, welches anzeigt, dass der Übertragungsvektor fehlerhaft empfangen wurde. +% Old Text +%Wenn wir den Übertragungsvektor jetzt Rücktransformieren wie im vorherigen Kapitel erhalten wir +%\[ +%r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]. +%\] +Jetzt stellt sich natürlich die Frage, wie wir daraus den ursprünglich gesendeten Nachrichtenvektor zurückerhalten sollen. Laut der Definition über die Funktionsweise eines Reed-Solomon-Codes können wir aus den Fehlerkorrekturstellen ein ``Lokatorpolynom'' berechnen, welches die Information enthält, welche stellen innerhalb des empfangenen Übertragungsvektors fehlerhaft sind. + +\subsection{Das Fehlerstellenpolynom $d(X)$ + \label{reedsolomon:subsection:fehlerpolynom}} +Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Möglichkeit finden, die Fehlerhaften von den Korrekten Stellen im Übertragungsvektor unterscheiden zu können. In einem ersten Versuch könnten wir $d$ berechnen mit +\begin{center} +\begin{tabular}{r c l} + $m(X)$ & $=$ & $4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \\ + $r(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ \\ + $d(X)$ & $=$ & $r(X) - m(X)$. +\end{tabular} +\end{center} +Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, wir werden werden deshalb erst in Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} darauf zurückkommen. + +Setzen wir jetzt noch unsere Einheitswurzel aus dem Beispiel ein so erhalten wir +% Old Text +%\begin{align} +% m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\ +% r(X) & = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7 \\ +% e(X) & = r(X) - m(X). +%\end{align} +%Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir +\begin{center} +\begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $d(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline +\end{tabular} +\end{center} +und damit die Information, dass allen Stellen, die nicht Null sind, Fehler enthalten. +Aus der Tabelle lesen wir, das in unserem Beispiel die Fehler an der Stelle drei und acht zu finden sind. + +Für das einfache Bestimmen von Hand mag dies ja noch ausreichen, jedoch können wir mit diesen Stellen nicht das Lokatorpolynom bestimmen, denn dafür bräuchten wir alle Nullstellen, an denen es Fehler gegeben hat (also sozusagen genau das umgekehrte). Um dies zu erreichen wenden wir eine andere Herangehensweise und nehmen uns den Satz von Fermat sowie den kleinsten gemeinsamen Teiler zur Hilfe. + +\subsection{Mit dem grössten gemeinsamen Teiler auf Nullstellenjagd +\label{reedsolomon:subsection:ggT}} + +Zuerst betrachten wir mal den Satz von Fermat deren Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass für +\[ +f(X) = X^{q-1} -1 = 0 +\] +wobei dies für jedes $q$ gilt. Setzen wir also das $q$ von unserem Beispiel ein +\[ +f(X) = X^{10}-1 = 0 \qquad \text{für } X = \{1,2,3,4,5,6,7,8,9,10\} +\] +und stellen dies als Nullstellenform (\textcolor{red}{richtiger name für die Schreibweise?}) dar. So ergibt sich die Darstellung +\[ +f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9). +\] +Zur Überprüfung können wir unsere Einheitswurzel in $a$ einsetzen und werden sehen, dass wir für $f(X) = 0$ erhalten werden. + +Wir können jetzt auch $d(X)$ nach der gleichen Überlegung darstellen als +\[ +d(X) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) \cdot p(x), +\] +wobei diese Darstellung nicht mehr alle Nullstellen umfasst wie es noch in $f(X)$ der Fall war. +Dies liegt daran, dass wir ja zwei Fehlerstellen (grau markiert) haben, die nicht Null sind. Diese fassen wir zum Restpolynom $p(X)$ (\textcolor{red}{eventuell farblich kennzeichnen?}) zusammen. +Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $d(X)$ berechnen, so erhalten wir mit +\[ +\operatorname{ggT}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) +\] +eine Liste von Nullstellen, an denen es keine Fehler gegeben hat. +Dies scheint zuerst nicht sehr hilfreich zu sein, da wir für das Lokatorpolynom ja eine Liste der Nullstellen suchen, an denen es Fehler gegeben hat. Aus diesem Grund berechnen wir im nächsten Schritt das kleinste gemeinsame Vielfache von $f(X)$ und $d(X)$. + +%Wir werden auch feststellen, das unsere Bemühungen bisher nicht umsonst waren. + +\subsection{Mit dem kgV fehlerhafte Nullstellen finden + \label{reedsolomon:subsection:kgV}} + +Das kgV hat nämlich die Eigenschaft sämtliche Nullstellen zu finden, also nicht nur die fehlerhaften sondern auch die korrekten, was in +\[ +\operatorname{kgV}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9) \cdot q(X). +\] +ersichtlich ist. +Aus dem vorherigen Abschnitt wissen wir auch, dass $d(X)$ alle korrekten Nullstellen beinhaltet. Teilen wir das kgV jetzt auf in +\[ +\operatorname{kgV}(f(X),d(X)) = d(X) \cdot l(X) +\] +sollten wir für $l(X)$ eine Liste mit allen fehlerhaften Nullstellen erhalten. +Somit ist +\[ +l(X) = (X-a^3)(X-a^8) +\] +unser gesuchtes Lokatorpolynom. +Es scheint so als müssten wir nur noch an den besagten Stellen den Übertragungsvektor korrigieren und wir währen fertig mit der Fehlerkorrektur. +Jedoch haben wir noch ein grundlegendes Problem, dass zu beginn aufgetaucht ist, wir aber beiseite geschoben haben. Die Rede ist natürlich vom Nachrichtenvektor $m(X)$, mit dem wir in erster Linie das wichtige Fehlerstellenpolynom $d(X)$ berechnet haben. + +\subsection{Der problematische Nachrichtenvektor $m(X)$ + \label{reedsolomon:subsection:nachrichtenvektor}} + +In Abschnitt \ref{reedsolomon:section:decmitfehler} haben wir +\[ +d(X) = r(X) - m(X) +\] +in Abhängigkeit von $m(X)$ berechnet. +Jedoch haben wir ausser acht gelassen, dass $m(X)$ auf der Empfängerseite nicht existiert und somit gänzlich unbekannt ist. +Es scheint so als würde dieser Lösungsansatz, den wir bisher verfolgt haben, nicht funktioniert. +Wir könnten uns höchstens noch fragen, ob wir tatsächlich nichts über den Nachrichtenvektor im Beispiel wissen. Wenn wir noch einmal den Vektor betrachten als +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +fällt uns aber auf, dass wir doch etwas über diesen Vektor wissen, nämlich den Wert der ersten 2t (im Beispiel vier) stellen. +Im Normalfall sollen diese nämlich den Wert null betragen und somit sind nur die letzten k stellen (im Beispiel sechs) für uns unbekannt, dargestellt als +\[ +m = [0,0,0,0,?,?,?,?,?,?]. +\] +Wie der Zufall es so will liegt an diesen vier Stellen auch die Information, wo die Fehlerstellen liegen. Daher reicht es auch aus +% darum werden die stellen auch als fehlerkorrekturstellen bezeichnet +\[ +d(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +\] +so zu berechnen, dass wir die wichtigen vier Stellen kennen, der Rest des Polynoms jedoch im unbekannten Restpolynom $p(X)$ enthalten ist. + +\textcolor{red}{ist das wechseln zwischen 2t,k aus dem allgemeinfall und vier,sechs aus dem beispiel zu verwirrend?} + +\subsection{Die Berechnung der Fehlerstellen + \label{reedsolomon:subsection:nachrichtenvektor}} + +Um die Fehlerstellen zu berechnen wenden wir die gleiche Vorgehensweise wie zuvor an, also zuerst den ggT, danach berechnen wir das kgV um am Ende das Lokatorpolynom zu erhalten. + +\subsubsection{Schritt 1: ggT} + +Wir berechnen den ggT von $f(X)$ und $d(X)$ mit +\begin{center} +\begin{tabular}{r c l} + $f(X)$ & $=$ & $X^{10} - 1 = X^{10} + 10$ \\ + $d(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ +\end{tabular} +\end{center} +% +% +% +%Das einzige Problem was jetzt noch bleibt ist, dass wir $e(X)$ berechnet haben aus +%\[ +%e(X) = r(X) - m(X), +%\] +%wobei $m(X)$ auf der Empfängerseite unbekannt ist. +%Es sieht danach aus, das wir diesen Lösungsansatz nicht verwenden können, da uns ein entscheidender Teil fehlt. +%Bei einer näheren Betrachtung von $m(X)$ fällt uns aber auf, dass wir doch etwas über $m(X)$ wissen. +%Wir kennen nämlich die ersten vier Stellen, da diese für die Fehlerkorrektur zuständig sind und daher Null sein müssen. +%\[ +%m = [0,0,0,0,?,?,?,?,?,?] +%\] +%An genau diesen Stellen liegt auch die Information, wo unsere Fehlerstellen liegen, was uns ermöglicht, den Teil von $e(X)$ zu berechnen, der uns auch interessiert. +% +%Wir können $e(X)$ also bestimmen als +%\[ +%e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +%\] +%wobei $p(X)$ wiederum ein unbekanntes Restpolynom ist und +%\[ +%f(X) = X^{10} - 1 = X^{10} + 10 +%\] +%ist können wir so in einer ersten Instanz den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen. +%Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berechnen so +% +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ +\end{array} +\] + +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ +\end{array} +\] +und erhalten +\[ +\operatorname{ggT}(f(X),e(X)) = 6X^8. +\] + +\subsubsection{Schritt 2: kgV} + +Mit dem Resultat das wir vom ggT erhalten haben können wir jetzt das kgV berechnen. Dazu können wir jetzt den erweiterten Euklidischen Algorithmus verwenden, den wir in Abschnitt \ref{buch:subsection:daskgv} kennengelernt haben. +% +%Mit den Resultaten, die wir vom Rechenweg des grössten gemeinsamen Teiler erhalten haben können wir jetzt auch das kleinste Gemeinsame Vielfache berechnen. Eine detailliertere Vorgehensweise findet man in Kapitel ???. +% +%Aus diesem erweiterten Euklidischen Algorithmus erhalten wir +\begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + +\end{center} +Daraus erhalten wir die Faktoren +\[ +l(X) = 2X^2 + 5 \qquad \rightarrow \qquad l(X) = 2(X-5)(X-6). +\] +Unser gesuchtes Lokatorpolynom hat also die Form +\[ +l(X) = (X-a^i)(X-a^j). +\] +Also brauchen wir nur noch $i$ und $j$ zu berechnen und wir haben unsere gesuchten Fehlerstellen. +Diese bekommen wir recht einfach mit +\begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^j = 6 \qquad \Rightarrow \qquad j = 8$. +\end{center} +Schlussendlich erhalten wir +\[ +d(X) = (X-a^3)(X-a^8) +\] +als unser Lokatorpolynom mit den fehlerhaften Stellen. diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex new file mode 100644 index 0000000..3b709f3 --- /dev/null +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -0,0 +1,116 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Decodierung: Ansatz ohne Fehler +\label{reedsolomon:section:decohnefehler}} +\rhead{fehlerlose rekonstruktion} + +In diesem Abschnitt betrachten wie die Überlegung, wie wir auf der Empfängerseite die Nachricht aus dem empfangenen Übertragungsvektor erhalten. Nach einer einfachen Überlegung müssen wir den Übertragungsvektor decodieren, was auf den ersten Blick nicht allzu kompliziert sein sollte, solange wir davon ausgehen können, dass es während der Übertragung keine Fehler gegeben hat. Wir betrachten deshalb den Übertragungskanal als fehlerfrei. + +Der Übertragungsvektor empfangen wir also als +\[ +v = [5,3,6,5,2,10,2,7,10,4]. +\] +% Old Text +%Im ersten Teil zur Decodierung des Übertragungsvektor betrachten wir den Übertragungskanal als fehlerfrei. +%Wir erhalten also unseren Übertragungsvektor +%\[ +%v = [5,3,6,5,2,10,2,7,10,4]. +%\] +Nach einem banalen Ansatz ist die Decodierung die Inverse der Codierung. Dank der Matrixschreibweise lässt sich dies relativ einfach umsetzen. +% Old Text +%Gesucht ist nun einen Weg, mit dem wir auf unseren Nachrichtenvektor zurückrechnen können. +%Ein banaler Ansatz ist das Invertieren der Glechung +\[ +v = A \cdot m \qquad \Rightarrow \qquad m = A^{-1} \cdot v +\] +Nur stellt sich jetzt die Frage, wie wir die Inverse von $A$ berechnen. +Dazu können wir wiederum den Ansatz der Fouriertransformation uns zur Hilfe nehmen, +jedoch betrachten wir jetzt deren Inverse. +Definiert ist sie als +\[ +F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \qquad \Rightarrow \qquad \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega. +\] +Damit beschäftigen wir uns im Abschnitt \ref{reedsolomon:subsection:algdec} weiter, konkret suchen wir momentan aber eine Inverse für unsere primitive Einheitswurzel $a$. +\[ +8^1 \qquad \rightarrow \qquad 8^{-1} +\] +Mit einem solchen Problem haben wir uns bereits in Abschnitt \ref{buch:section:euklid} befasst und so den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. + +% Old Text +%Im Abschnitt \textcolor{red}{4.1} haben wir den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. + +\subsection{Inverse der primitiven Einheitswurzel +\label{reedsolomon:subsection:invEinh}} + +Die Funktionsweise des euklidischen Algorithmus ist im Kapitel \ref{buch:section:euklid} ausführlich beschrieben. +Für unsere Anwendung wählen wir die Parameter $a = 8$ und $b = 11$ ($\mathbb{F}_{11}$). +Daraus erhalten wir + +\begin{center} + +\begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline +\end{tabular} + +\end{center} +\begin{center} + +\begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + +\end{tabular} + +\end{center} +als Inverse der primitiven Einheitswurzel. Die inverse Transformationsmatrix $A^{-1}$ bilden wir indem wir jetzt die inverse primitive Einheitswurzel anstelle der primitiven Einheitswurzel in die Matrix einsetzen. + +\subsection{Allgemeine Decodierung + \label{reedsolomon:subsection:algdec}} + +Wir haben jetzt fast alles für eine erfolgreiche Rücktransformation beisammen. Wir haben aber noch nicht alle Aspekte der inversen diskreten Fouriertransformation befolgt, so fehlt uns noch einen Vorfaktor +\[ +m = \textcolor{red}{s} \cdot A^{-1} \cdot v +\] +den wir noch bestimmen müssen. +Glücklicherweise lässt der sich analog wie bei der inversen diskreten Fouriertransformation bestimmen und beträgt +\[ +s = \frac{1}{10}. +\] +Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. Somit lässt sich der Nachrichtenvektor einfach bestimmen mit +\[ +m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ +\end{pmatrix} +\] +und wir erhalten +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +als unsere Nachricht zurück.
\ No newline at end of file diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex new file mode 100644 index 0000000..146067a --- /dev/null +++ b/buch/papers/reedsolomon/endlichekoerper.tex @@ -0,0 +1,23 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Reed-Solomon in Endlichen Körpern +\label{reedsolomon:section:endlichekoerper}} +\rhead{Problemstellung} + +\textcolor{red}{TODO: (warten auf den 1. Teil)} + +Das Rechnen in endlichen Körpern bietet einige Vorteile: + +\begin{itemize} + \item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten. + + \item Digitale Fehlerkorrektur: lässt sich nur in endlichen Körpern umsetzen. + +\end{itemize} + +Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen. + +Nun stellt sich die Frage, wie wir eine fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes ``Lokatorpolynom'' generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. diff --git a/buch/papers/reedsolomon/experiments/f.m b/buch/papers/reedsolomon/experiments/f.m new file mode 100644 index 0000000..6bdc741 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/f.m @@ -0,0 +1,61 @@ +# +# f.m -- Reed-Solomon-Visualisierung mit FFT +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +N = 64; +b = 32; +l = N + b; + +signal = zeros(l,1); +signal(1:N,1) = round(10 * rand(N,1)); +signal + +plot(abs(signal)); +xlim([1, l]); +title("Signal"); +pause() + +codiert = fft(signal) + +plot(abs(codiert)); +xlim([1, l]); +title("Codiert"); +pause() + +fehler = zeros(l,1); +fehler(21,1) = 2; +fehler(75,1) = 1; +fehler(7,1) = 2; + +plot(fehler); +xlim([1, l]); +title("Fehler"); +pause() + +empfangen = codiert + fehler; + +plot(abs(empfangen)); +xlim([1, l]); +title("Empfangen"); +pause() + +decodiert = ifft(empfangen) +plot(abs(decodiert)); +xlim([1, l]); +title("Decodiert"); +pause() + +syndrom = decodiert; +syndrom(1:N,1) = zeros(N,1) +plot(abs(syndrom)); +xlim([1, l]); +title("Syndrom"); +pause() + +locator = abs(fft(syndrom)) + +plot(locator); +xlim([1, l]); +title("Locator"); +pause() diff --git a/buch/papers/reedsolomon/hilfstabellen.tex b/buch/papers/reedsolomon/hilfstabellen.tex new file mode 100644 index 0000000..4e39de5 --- /dev/null +++ b/buch/papers/reedsolomon/hilfstabellen.tex @@ -0,0 +1,19 @@ +% +% hilfstabellen.tex +% Autor: Michael Steiner +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{$\mathbb{F}_{11}$ Hilfstabellen + \label{reedsolomon:section:hilfstabellen}} +\rhead{Hilfstabellen} + +Um das rechnen zu erleichtern findet man in diesem Abschnitt die Resultate, die bei der Addition und der Multiplikation in $\mathbb{F}_{11}$ resultieren. + +\subsection{Additionstabelle + \label{reedsolomon:subsection:adtab}} +\input{papers/reedsolomon/restetabelle1.tex} + +\subsection{Multiplikationstabelle + \label{reedsolomon:subsection:mptab}} +\input{papers/reedsolomon/restetabelle2.tex}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index c04be7f..fa20936 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -3,10 +3,10 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:reedsolomon}} +\chapter{Reed-Solomon-Code\label{chapter:reedsolomon}} \lhead{Thema} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Joshua Bär und Michael Steiner} Ein paar Hinweise für die korrekte Formatierung des Textes \begin{itemize} @@ -27,10 +27,25 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. \end{itemize} +% Joshua \input{papers/reedsolomon/teil0.tex} \input{papers/reedsolomon/teil1.tex} \input{papers/reedsolomon/teil2.tex} \input{papers/reedsolomon/teil3.tex} +% Michael +\input{papers/reedsolomon/endlichekoerper} +\input{papers/reedsolomon/codebsp} +\input{papers/reedsolomon/decohnefehler} +\input{papers/reedsolomon/decmitfehler} +\input{papers/reedsolomon/rekonstruktion} +\input{papers/reedsolomon/nachschlagewerk} +\input{papers/reedsolomon/hilfstabellen} +%\input{papers/reedsolomon/anwendungen} -> geplant + +\nocite{reedsolomon:weitz} +\nocite{reedsolomon:informationkommunikation} +%\nocite{reedsolomon:mendezmueller} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/reedsolomon/nachschlagewerk.tex b/buch/papers/reedsolomon/nachschlagewerk.tex new file mode 100644 index 0000000..60b857e --- /dev/null +++ b/buch/papers/reedsolomon/nachschlagewerk.tex @@ -0,0 +1,4 @@ +\section{Nachschlagewerk + \label{reedsolomon:section:nachschlagen}} +\rhead{nachschlagewerk} +todo: auflistung von z.b nachrichtenvektor, übertragungsvektor usw. inklusiver erklärung was es ist falls man beim lesen den faden verliert
\ No newline at end of file diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib index 38613bd..731bd35 100644 --- a/buch/papers/reedsolomon/references.bib +++ b/buch/papers/reedsolomon/references.bib @@ -4,32 +4,22 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{reedsolomon:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@online{reedsolomon:weitz, + title = {Fehlerkorrektur mit Reed-Solomon-Codes}, + url = {https://youtu.be/uOLW43OIZJ0}, + date = {2021-06-10}, + year = {2021}, + month = {6}, + day = {10} } -@book{reedsolomon:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{reedsolomon:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@book{reedsolomon:informationkommunikation, + title = {Information und Kommunikation}, + author = {Markus Hufschmid}, + publisher = {Teubner}, + year = {2007}, + isbn = {978-3-8351-0122-7}, + inseries = {}, + volume = {1} } diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex new file mode 100644 index 0000000..89a700f --- /dev/null +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -0,0 +1,185 @@ +% +% rekonstruktion.tex +% Autor: Michael Steiner +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Nachricht Rekonstruieren +\label{reedsolomon:section:rekonstruktion}} +\rhead{Rekonstruktion} +Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können. +Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren. +Das Lokatorpolynom +\[ +l(X) = (X - a^3)(X-a^8) +\] +markiert dabei diese fehlerhaften Stellen im Übertragungsvektor +\[ +w = [5,3,6,8,2,10,2,7,1,4]. +\] +Als Ausgangslage verwenden wir die Matrix, mit der wir den Nachrichtenvektor ursprünglich codiert haben. +Unser Ziel ist es wie auch schon im Abschnitt \ref{reedsolomon:section:decohnefehler} eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. +Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Abschnitt \ref{reedsolomon:section:decohnefehler} angewendet haben. + +Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen. +\[ +\textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ +\end{pmatrix}} +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ +\end{pmatrix} +\] +Die rot markierten Stellen im Übertragungsvektor enthalten Fehler und bringt uns daher kein weiterer Nutzen. +Aus diesem Grund werden diese Stellen aus dem Vektor entfernt, was wir hier ohne Probleme machen können, da dieser Code ja über Fehlerkorrekturstellen verfügt, deren Aufgabe es ist, eine bestimmte Anzahl an Fehler kompensieren zu können. +Die dazugehörigen Zeilen in der Matrix werden ebenfalls entfernt, da die Matrix gleich viele Zeilen wie im Übertragungsvektor aufweisen muss, damit man ihn decodieren kann. + +Daraus resultiert +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ +\end{pmatrix} +. +\] +Die Matrix ist jedoch nicht mehr quadratisch, was eine Rekonstruktion durch Inversion ausschliesst. +Um die quadratische Form wieder herzustellen müssen wir zwei Spalten aus der Matrix entfernen. +Wir kennen aber das Resultat aus den letzten vier Spalten, da wir wissen, das die Nachricht aus Nutzdatenteil und Fehlerkorrekturteil besteht, wobei der letzteres bekanntlich aus lauter Nullstellen besteht. +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{darkgreen}{8^6}& \textcolor{darkgreen}{8^7}& \textcolor{darkgreen}{8^8}& \textcolor{darkgreen}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{darkgreen}{8^{12}}& \textcolor{darkgreen}{8^{14}}& \textcolor{darkgreen}{8^{16}}& \textcolor{darkgreen}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{darkgreen}{8^{24}}& \textcolor{darkgreen}{8^{28}}& \textcolor{darkgreen}{8^{32}}& \textcolor{darkgreen}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{darkgreen}{8^{30}}& \textcolor{darkgreen}{8^{35}}& \textcolor{darkgreen}{8^{40}}& \textcolor{darkgreen}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{darkgreen}{8^{36}}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{48}}& \textcolor{darkgreen}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{49}}& \textcolor{darkgreen}{8^{56}}& \textcolor{darkgreen}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{darkgreen}{8^{54}}& \textcolor{darkgreen}{8^{63}}& \textcolor{darkgreen}{8^{72}}& \textcolor{darkgreen}{8^{81}}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{darkgreen}{m_6} \\ \textcolor{darkgreen}{m_7} \\ \textcolor{darkgreen}{m_8} \\ \textcolor{darkgreen}{m_9} \\ +\end{pmatrix} +\] +Wir nehmen die entsprechenden Spalten aus der Matrix heraus und erhalten so das Überbestimmte Gleichungssystem +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +. +\] +Die roten Zeilen können wir aufgrund der Überbestimmtheit ebenfalls entfernen und erhalten so die gesuchte quadratische Matrix +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +. +\] +Nun können wir den Gauss-Algorithmus anwenden um die Matrix zu Invertieren. +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} += +\begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +\qquad +\Rightarrow +\qquad +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} += +\begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} +\] +Multiplizieren wir nun aus, erhalten wir unseren Nutzdatenteil +\[ +m = [4,7,2,5,8,1] +\] +zurück, den wir ursprünglich versendet haben. + diff --git a/buch/papers/reedsolomon/restetabelle1.tex b/buch/papers/reedsolomon/restetabelle1.tex new file mode 100644 index 0000000..3969ef2 --- /dev/null +++ b/buch/papers/reedsolomon/restetabelle1.tex @@ -0,0 +1,176 @@ +% created by Michael Steiner +% +% Restetabelle von F_11: Addition + +% alternatives design +%\begin{figure} +%\begin{center} +%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +%\hline +%+&0&1&2&3&4&5&6&7&8&9&10\\ +%\hline +%0&0&1&2&3&4&5&6&7&8&9&10\\ +%1&1&2&3&4&5&6&7&8&9&10&0\\ +%2&2&3&4&5&6&7&8&9&10&0&1\\ +%3&3&4&5&6&7&8&9&10&0&1&2\\ +%4&4&5&6&7&8&9&10&0&1&2&3\\ +%5&5&6&7&8&9&10&0&1&2&3&4\\ +%6&6&7&8&9&10&0&1&2&3&4&5\\ +%7&7&8&9&10&0&1&2&3&4&5&6\\ +%8&8&9&10&0&1&2&3&4&5&6&7\\ +%9&9&10&0&1&2&3&4&5&6&7&8\\ +%10&10&0&1&2&3&4&5&6&7&8&9\\ +%\hline +%\end{tabular} +%\end{center} +%\end{figure} + +\begin{center} + +\begin{tikzpicture}[>=latex,thick,scale=0.45] +\fill[color=gray!40] (0,0) rectangle (18,-1.5); +\fill[color=gray!40] (0,0) rectangle (1.5,-18); +\draw[step = 1.5, gray,very thin] (0,0) grid (18,-18); +\draw[very thick] (0,0) rectangle (18,-18); +\draw[very thick] (0,-1.5) -- (18,-1.5); +\draw[very thick] (1.5,0) -- (1.5,-18); +\node at (0.75,-0.75) {$+$}; +\foreach \x in {0,...,10} + \node at (2.25+\x*1.5,-0.75) {$\x$}; +\foreach \y in {0,...,10} + \node at (0.75,-2.25+\y*-1.5) {$\y$}; +% Row 0 +\node at ( 2.25,-2.25) {$0$}; +\node at ( 3.75,-2.25) {$1$}; +\node at ( 5.25,-2.25) {$2$}; +\node at ( 6.75,-2.25) {$3$}; +\node at ( 8.25,-2.25) {$4$}; +\node at ( 9.75,-2.25) {$5$}; +\node at (11.25,-2.25) {$6$}; +\node at (12.75,-2.25) {$7$}; +\node at (14.25,-2.25) {$8$}; +\node at (15.75,-2.25) {$9$}; +\node at (17.25,-2.25) {$10$}; +% Row 1 +\node at ( 2.25,-3.75) {$1$}; +\node at ( 3.75,-3.75) {$2$}; +\node at ( 5.25,-3.75) {$3$}; +\node at ( 6.75,-3.75) {$4$}; +\node at ( 8.25,-3.75) {$5$}; +\node at ( 9.75,-3.75) {$6$}; +\node at (11.25,-3.75) {$7$}; +\node at (12.75,-3.75) {$8$}; +\node at (14.25,-3.75) {$9$}; +\node at (15.75,-3.75) {$10$}; +\node at (17.25,-3.75) {$0$}; +% Row 2 +\node at ( 2.25,-5.25) {$2$}; +\node at ( 3.75,-5.25) {$3$}; +\node at ( 5.25,-5.25) {$4$}; +\node at ( 6.75,-5.25) {$5$}; +\node at ( 8.25,-5.25) {$6$}; +\node at ( 9.75,-5.25) {$7$}; +\node at (11.25,-5.25) {$8$}; +\node at (12.75,-5.25) {$9$}; +\node at (14.25,-5.25) {$10$}; +\node at (15.75,-5.25) {$0$}; +\node at (17.25,-5.25) {$1$}; +% Row 3 +\node at ( 2.25,-6.75) {$3$}; +\node at ( 3.75,-6.75) {$4$}; +\node at ( 5.25,-6.75) {$5$}; +\node at ( 6.75,-6.75) {$6$}; +\node at ( 8.25,-6.75) {$7$}; +\node at ( 9.75,-6.75) {$8$}; +\node at (11.25,-6.75) {$9$}; +\node at (12.75,-6.75) {$10$}; +\node at (14.25,-6.75) {$0$}; +\node at (15.75,-6.75) {$1$}; +\node at (17.25,-6.75) {$2$}; +% Row 4 +\node at ( 2.25,-8.25) {$4$}; +\node at ( 3.75,-8.25) {$5$}; +\node at ( 5.25,-8.25) {$6$}; +\node at ( 6.75,-8.25) {$7$}; +\node at ( 8.25,-8.25) {$8$}; +\node at ( 9.75,-8.25) {$9$}; +\node at (11.25,-8.25) {$10$}; +\node at (12.75,-8.25) {$0$}; +\node at (14.25,-8.25) {$1$}; +\node at (15.75,-8.25) {$2$}; +\node at (17.25,-8.25) {$3$}; +% Row 5 +\node at ( 2.25,-9.75) {$5$}; +\node at ( 3.75,-9.75) {$6$}; +\node at ( 5.25,-9.75) {$7$}; +\node at ( 6.75,-9.75) {$8$}; +\node at ( 8.25,-9.75) {$9$}; +\node at ( 9.75,-9.75) {$10$}; +\node at (11.25,-9.75) {$0$}; +\node at (12.75,-9.75) {$1$}; +\node at (14.25,-9.75) {$2$}; +\node at (15.75,-9.75) {$3$}; +\node at (17.25,-9.75) {$4$}; +% Row 6 +\node at ( 2.25,-11.25) {$6$}; +\node at ( 3.75,-11.25) {$7$}; +\node at ( 5.25,-11.25) {$8$}; +\node at ( 6.75,-11.25) {$9$}; +\node at ( 8.25,-11.25) {$10$}; +\node at ( 9.75,-11.25) {$0$}; +\node at (11.25,-11.25) {$1$}; +\node at (12.75,-11.25) {$2$}; +\node at (14.25,-11.25) {$3$}; +\node at (15.75,-11.25) {$4$}; +\node at (17.25,-11.25) {$5$}; +% Row 7 +\node at ( 2.25,-12.75) {$7$}; +\node at ( 3.75,-12.75) {$8$}; +\node at ( 5.25,-12.75) {$9$}; +\node at ( 6.75,-12.75) {$10$}; +\node at ( 8.25,-12.75) {$0$}; +\node at ( 9.75,-12.75) {$1$}; +\node at (11.25,-12.75) {$2$}; +\node at (12.75,-12.75) {$3$}; +\node at (14.25,-12.75) {$4$}; +\node at (15.75,-12.75) {$5$}; +\node at (17.25,-12.75) {$6$}; +% Row 8 +\node at ( 2.25,-14.25) {$8$}; +\node at ( 3.75,-14.25) {$9$}; +\node at ( 5.25,-14.25) {$10$}; +\node at ( 6.75,-14.25) {$0$}; +\node at ( 8.25,-14.25) {$1$}; +\node at ( 9.75,-14.25) {$2$}; +\node at (11.25,-14.25) {$3$}; +\node at (12.75,-14.25) {$4$}; +\node at (14.25,-14.25) {$5$}; +\node at (15.75,-14.25) {$6$}; +\node at (17.25,-14.25) {$7$}; +% Row 9 +\node at ( 2.25,-15.75) {$9$}; +\node at ( 3.75,-15.75) {$10$}; +\node at ( 5.25,-15.75) {$0$}; +\node at ( 6.75,-15.75) {$1$}; +\node at ( 8.25,-15.75) {$2$}; +\node at ( 9.75,-15.75) {$3$}; +\node at (11.25,-15.75) {$4$}; +\node at (12.75,-15.75) {$5$}; +\node at (14.25,-15.75) {$6$}; +\node at (15.75,-15.75) {$7$}; +\node at (17.25,-15.75) {$8$}; +% Row 10 +\node at ( 2.25,-17.25) {$10$}; +\node at ( 3.75,-17.25) {$0$}; +\node at ( 5.25,-17.25) {$1$}; +\node at ( 6.75,-17.25) {$2$}; +\node at ( 8.25,-17.25) {$3$}; +\node at ( 9.75,-17.25) {$4$}; +\node at (11.25,-17.25) {$5$}; +\node at (12.75,-17.25) {$6$}; +\node at (14.25,-17.25) {$7$}; +\node at (15.75,-17.25) {$8$}; +\node at (17.25,-17.25) {$9$}; +\end{tikzpicture} + +\end{center} diff --git a/buch/papers/reedsolomon/restetabelle2.tex b/buch/papers/reedsolomon/restetabelle2.tex new file mode 100644 index 0000000..1a9815c --- /dev/null +++ b/buch/papers/reedsolomon/restetabelle2.tex @@ -0,0 +1,176 @@ +% created by Michael Steiner +% +% Restetabelle von F_11: Multiplikation + +% alternatives design +%\begin{figure} +%\begin{center} +%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +%\hline +%\cdot&0&1&2&3&4&5&6&7&8&9&10\\ +%\hline +%0&0&0&0&0&0&0&0&0&0&0&0\\ +%1&0&1&2&3&4&5&6&7&8&9&10\\ +%2&0&2&4&6&8&10&1&3&5&7&9\\ +%3&0&3&6&9&1&4&7&10&2&5&8\\ +%4&0&4&8&1&5&9&2&6&10&3&7\\ +%5&0&5&10&4&9&3&8&2&7&1&6\\ +%6&0&6&1&7&2&8&3&9&4&10&5\\ +%7&0&7&3&10&6&2&9&5&1&8&4\\ +%8&0&8&5&2&10&7&4&1&9&6&3\\ +%9&0&9&7&5&3&1&10&8&6&4&2\\ +%10&0&10&9&8&7&6&5&4&3&2&1\\ +%\hline +%\end{tabular} +%\end{center} +%\end{figure} + +\begin{center} + + \begin{tikzpicture}[>=latex,thick,scale=0.45] + \fill[color=gray!40] (0,0) rectangle (18,-1.5); + \fill[color=gray!40] (0,0) rectangle (1.5,-18); + \draw[step = 1.5, gray,very thin] (0,0) grid (18,-18); + \draw[very thick] (0,0) rectangle (18,-18); + \draw[very thick] (0,-1.5) -- (18,-1.5); + \draw[very thick] (1.5,0) -- (1.5,-18); + \node at (0.75,-0.75) {$\cdot$}; + \foreach \x in {0,...,10} + \node at (2.25+\x*1.5,-0.75) {$\x$}; + \foreach \y in {0,...,10} + \node at (0.75,-2.25+\y*-1.5) {$\y$}; + % Row 0 + \node at ( 2.25,-2.25) {$0$}; + \node at ( 3.75,-2.25) {$0$}; + \node at ( 5.25,-2.25) {$0$}; + \node at ( 6.75,-2.25) {$0$}; + \node at ( 8.25,-2.25) {$0$}; + \node at ( 9.75,-2.25) {$0$}; + \node at (11.25,-2.25) {$0$}; + \node at (12.75,-2.25) {$0$}; + \node at (14.25,-2.25) {$0$}; + \node at (15.75,-2.25) {$0$}; + \node at (17.25,-2.25) {$0$}; + % Row 1 + \node at ( 2.25,-3.75) {$0$}; + \node at ( 3.75,-3.75) {$1$}; + \node at ( 5.25,-3.75) {$2$}; + \node at ( 6.75,-3.75) {$3$}; + \node at ( 8.25,-3.75) {$4$}; + \node at ( 9.75,-3.75) {$5$}; + \node at (11.25,-3.75) {$6$}; + \node at (12.75,-3.75) {$7$}; + \node at (14.25,-3.75) {$8$}; + \node at (15.75,-3.75) {$9$}; + \node at (17.25,-3.75) {$10$}; + % Row 2 + \node at ( 2.25,-5.25) {$0$}; + \node at ( 3.75,-5.25) {$2$}; + \node at ( 5.25,-5.25) {$4$}; + \node at ( 6.75,-5.25) {$6$}; + \node at ( 8.25,-5.25) {$8$}; + \node at ( 9.75,-5.25) {$10$}; + \node at (11.25,-5.25) {$1$}; + \node at (12.75,-5.25) {$3$}; + \node at (14.25,-5.25) {$5$}; + \node at (15.75,-5.25) {$7$}; + \node at (17.25,-5.25) {$9$}; + % Row 3 + \node at ( 2.25,-6.75) {$0$}; + \node at ( 3.75,-6.75) {$3$}; + \node at ( 5.25,-6.75) {$6$}; + \node at ( 6.75,-6.75) {$9$}; + \node at ( 8.25,-6.75) {$1$}; + \node at ( 9.75,-6.75) {$4$}; + \node at (11.25,-6.75) {$7$}; + \node at (12.75,-6.75) {$10$}; + \node at (14.25,-6.75) {$2$}; + \node at (15.75,-6.75) {$5$}; + \node at (17.25,-6.75) {$8$}; + % Row 4 + \node at ( 2.25,-8.25) {$0$}; + \node at ( 3.75,-8.25) {$4$}; + \node at ( 5.25,-8.25) {$8$}; + \node at ( 6.75,-8.25) {$1$}; + \node at ( 8.25,-8.25) {$5$}; + \node at ( 9.75,-8.25) {$9$}; + \node at (11.25,-8.25) {$2$}; + \node at (12.75,-8.25) {$6$}; + \node at (14.25,-8.25) {$10$}; + \node at (15.75,-8.25) {$3$}; + \node at (17.25,-8.25) {$7$}; + % Row 5 + \node at ( 2.25,-9.75) {$0$}; + \node at ( 3.75,-9.75) {$5$}; + \node at ( 5.25,-9.75) {$10$}; + \node at ( 6.75,-9.75) {$4$}; + \node at ( 8.25,-9.75) {$9$}; + \node at ( 9.75,-9.75) {$3$}; + \node at (11.25,-9.75) {$8$}; + \node at (12.75,-9.75) {$2$}; + \node at (14.25,-9.75) {$7$}; + \node at (15.75,-9.75) {$1$}; + \node at (17.25,-9.75) {$6$}; + % Row 6 + \node at ( 2.25,-11.25) {$0$}; + \node at ( 3.75,-11.25) {$6$}; + \node at ( 5.25,-11.25) {$1$}; + \node at ( 6.75,-11.25) {$7$}; + \node at ( 8.25,-11.25) {$2$}; + \node at ( 9.75,-11.25) {$8$}; + \node at (11.25,-11.25) {$3$}; + \node at (12.75,-11.25) {$9$}; + \node at (14.25,-11.25) {$4$}; + \node at (15.75,-11.25) {$10$}; + \node at (17.25,-11.25) {$5$}; + % Row 7 + \node at ( 2.25,-12.75) {$0$}; + \node at ( 3.75,-12.75) {$7$}; + \node at ( 5.25,-12.75) {$3$}; + \node at ( 6.75,-12.75) {$10$}; + \node at ( 8.25,-12.75) {$6$}; + \node at ( 9.75,-12.75) {$2$}; + \node at (11.25,-12.75) {$9$}; + \node at (12.75,-12.75) {$5$}; + \node at (14.25,-12.75) {$1$}; + \node at (15.75,-12.75) {$8$}; + \node at (17.25,-12.75) {$4$}; + % Row 8 + \node at ( 2.25,-14.25) {$0$}; + \node at ( 3.75,-14.25) {$8$}; + \node at ( 5.25,-14.25) {$5$}; + \node at ( 6.75,-14.25) {$2$}; + \node at ( 8.25,-14.25) {$10$}; + \node at ( 9.75,-14.25) {$7$}; + \node at (11.25,-14.25) {$4$}; + \node at (12.75,-14.25) {$1$}; + \node at (14.25,-14.25) {$9$}; + \node at (15.75,-14.25) {$6$}; + \node at (17.25,-14.25) {$3$}; + % Row 9 + \node at ( 2.25,-15.75) {$0$}; + \node at ( 3.75,-15.75) {$9$}; + \node at ( 5.25,-15.75) {$7$}; + \node at ( 6.75,-15.75) {$5$}; + \node at ( 8.25,-15.75) {$3$}; + \node at ( 9.75,-15.75) {$1$}; + \node at (11.25,-15.75) {$10$}; + \node at (12.75,-15.75) {$8$}; + \node at (14.25,-15.75) {$6$}; + \node at (15.75,-15.75) {$4$}; + \node at (17.25,-15.75) {$2$}; + % Row 10 + \node at ( 2.25,-17.25) {$0$}; + \node at ( 3.75,-17.25) {$10$}; + \node at ( 5.25,-17.25) {$9$}; + \node at ( 6.75,-17.25) {$8$}; + \node at ( 8.25,-17.25) {$7$}; + \node at ( 9.75,-17.25) {$6$}; + \node at (11.25,-17.25) {$5$}; + \node at (12.75,-17.25) {$4$}; + \node at (14.25,-17.25) {$3$}; + \node at (15.75,-17.25) {$2$}; + \node at (17.25,-17.25) {$1$}; + \end{tikzpicture} + +\end{center}
\ No newline at end of file |
