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Merge branch 'master' of bernard.com.de:mysaa/arbeiten/TIPE2021

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Mysaa 2021-07-10 21:51:36 +02:00
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CompteRendu/.gitignore vendored
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*.toc
*.thm
svg-inkscape/
print_output_path.svg
*_path.svg
*.bbl
*.bcf
*.blg
*.run.xml
*.nav
*.out
*.snm

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% !TeX encoding = UTF-8
\documentclass[10pt,a4paper]{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{csquotes}
\usepackage[french]{babel}
\usepackage{tabularx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{calrsfs}
\usepackage{setspace}
\usepackage{stmaryrd}
\usepackage{svg}
\usepackage{bashful}
\usepackage{pdfpages}
\uselanguage{French}
\languagepath{French}
\lstset{
numbers=left,
breaklines=true,
tabsize=2,
basicstyle=\ttfamily,
}
% Racourcis
\newcommand{\FD}{\mathbb{F}_2}
\newcommand\fsign[3]{\item \textbf{#1} : \textit{#2} \hfill Complexité~en~$\mathcal{O}(#3)$\\}
\newcommand\fsignz[2]{\item \textbf{#1} : \textit{#2}\\}
\newcommand{\into}{\textrightarrow\:}
%\newtheorem{definition}{Définition}[section]
%\newtheorem{theoreme}{Théorème}[section]
%\theoremstyle{remark}
\newtheorem*{remarque}{Remarque}
\renewcommand\unskip\relax
\usetheme{Madrid}
%\hypersetup{pdfpagemode=FullScreen}
% Transition en fade-in par défaut
%\addtobeamertemplate{background canvas}{\transfade[duration=0.4]}{}
\author{Samy AVRILLON}
\title{Étude des codes correcteur d'erreurs}
\begin{document}
\begin{frame}
\maketitle
\end{frame}
\section*{Introduction}
\begin{frame}
\frametitle{Introduction}
\pause
\begin{figure}[h]
\begin{center}
\includegraphics[width=.2\linewidth]{gaussV0-llow.png}
\includegraphics[width=.2\linewidth]{gaussVH-llow.png}
\hspace{.1\linewidth}
\includegraphics[width=.2\linewidth]{gaussV0-low.png}
\includegraphics[width=.2\linewidth]{gaussVH-low.png}
\small
\end{center}
\caption{Exemple d'application de codes correcteurs}
\label{gaussEx}
\end{figure}
\end{frame}
\section*{Sommaire}
\begin{frame}
\frametitle{Sommaire}
\pause
\tableofcontents[pausesections]
\end{frame}
\addcontentsline{toc}{section}{Principe}
\begin{frame}
\frametitle{Transmission simple}
\includesvg[width=0.98\textwidth]{diagramme_transmission_simplet_path.svg}
\end{frame}
\begin{frame}
\frametitle{Transmission avec code correcteur}
\includesvg[width=0.98\textwidth]{diagramme_transmission_corrige_path.svg}
\end{frame}
\begin{frame}
\centering\Huge Contexte mathématique
\end{frame}
\section{Contexte mathématique}
\subsection{Codes}
\begin{frame}
\frametitle{Codes sur $\FD$}
\begin{definition}[Code, Codage]
\pause
Code $(n,k)$: partie de $\FD^n$
\pause
Codage: application $\FD^k \rightarrow \FD^n$ injective.
\pause
Codage systématique
\end{definition}
\pause
\begin{definition}[Distance de Hamming, Poids d'un mot]
$$ d(v,w) = \operatorname{card}(i\in \llbracket1,n\rrbracket,v_i \neq w_i)$$
\pause
$$ w(v) = d(v,0) = \operatorname{card}(i\in \llbracket1,n\rrbracket,v_i = 0)$$
\end{definition}
\end{frame}
\begin{frame}
\[
\scalebox{1.7}{
$
\begin{bmatrix}
1\\0\\0\\1\\1\\0\\1\\1\\0\\0\\1
\end{bmatrix}
\begin{bmatrix}
0\\1\\0\\1\\0\\0\\1\\1\\1\\1\\1
\end{bmatrix}
$
}\]
\end{frame}
\begin{frame}
\[
\scalebox{1.7}{
$
\begin{bmatrix}
\color{blue}1\\\color{blue}0\\0\\1\\\color{blue}1\\0\\1\\1\\\color{blue}0\\\color{blue}0\\1
\end{bmatrix}
\begin{bmatrix}
\color{blue}0\\\color{blue}1\\0\\1\\\color{blue}0\\0\\1\\1\\\color{blue}1\\\color{blue}1\\1
\end{bmatrix}
$
}\]
\end{frame}
\begin{frame}
\[
\scalebox{1.7}{
$
\begin{bmatrix}
\color{red}1\\0\\0\\\color{red}1\\\color{red}1\\0\\\color{red}1\\\color{red}1\\0\\0\\\color{red}1
\end{bmatrix}
\begin{bmatrix}
0\\\color{red}1\\0\\\color{red}1\\0\\0\\\color{red}1\\\color{red}1\\\color{red}1\\\color{red}1\\\color{red}1
\end{bmatrix}
$
}\]
\end{frame}
\begin{frame}
\frametitle{Propriétés des codes}
\begin{definition}
\pause
Capacité de détection $e_d$ et capacité de correction $e_c$.
\pause
La distance minimale.
$$d_C = \min_{x,y\in C\times C,x\neq y}\left(d(x,y)\right)$$
\end{definition}
\end{frame}
\begin{frame}
\frametitle{Distance minimale}
\begin{theorem}[Distance minimale]
\pause
$$ e_d = d_C - 1 \qquad\pause e_c = \left\lfloor\frac{d_C - 1}{2}\right\rfloor$$
\end{theorem}
\end{frame}
\begin{frame}
\frametitle{Code parfait}
\begin{definition}[Code Parfait]
\pause
Jamais d'ambigüité sur la façon de décoder un mot érroné.
\end{definition}
\pause
\textit{Exemple: Code de répétition (n,1)}
$$C = \left\{\begin{bmatrix}
0\\\vdots\\0
\end{bmatrix},
\begin{bmatrix}
1\\\vdots\\1
\end{bmatrix}\right\}$$
\pause
$$d_C = n\pause,e_d = n-1\pause,e_c = \left\lfloor \frac{n-1}{2} \right\rfloor$$
\pause\centering
parfait \underline{si et seulement si} $n$ impair
\end{frame}
\subsection{Codes linéaires}
\begin{frame}
\frametitle{Codes linéaire}
\begin{definition}
\pause
$C$ sous-espace vectoriel de $\FD^n$
\pause
Codage linéaire
\end{definition}
\end{frame}
\begin{frame}
\frametitle{Matrices du code linéaire}
\begin{definition}[Matrice génératrice]
\pause
$$G \in \mathcal{M}_{n,k}(\FD)$$
$$Im(G) = C$$
\end{definition}
\pause
\begin{definition}[Matrice de contrôle]
$$H \in \mathcal{M}_{n-k,n}(\FD)$$
$$H\cdot X = 0 \quad\text{\underline{ssi}}\quad X \in C$$
\end{definition}
\end{frame}
\begin{frame}
\frametitle{Calcul de distances}
\begin{theorem}[Calcul de distances]
\pause
$$d_C = \min_{X\in C, X\neq 0}(w(X))$$
\pause
Borne de Singleton:
$$d_C \leqslant n+1-k$$
\end{theorem}
\pause
\begin{definition}[Erreur et syndrome]
\pause
$$E = Z' - Y'$$
\pause
$$S(Z) = H \cdot Z$$
\pause
$$S(E) = S(Z)$$
\end{definition}
\end{frame}
\subsection{Codes cycliques}
\begin{frame}
\frametitle{Codes cycliques}
\begin{definition}[Code cyclique]
\pause
$w_1w_2w_3\ldots w_n \in C \implies w_2w_3w_4\ldots w_{n-1}w_nw_1 \in C$.
\end{definition}
\pause
\begin{definition}[Mot binaire associé à un polynôme]
\pause
\[
\begin{bmatrix}
1\\0\\0\\1\\1\\0\\1\\1\\0\\0\\1
\end{bmatrix}\mapsto 1 + X^3+X^4+X^6 + X^7 + X^{10} \]
\end{definition}
\end{frame}
\begin{frame}
\frametitle{Des polynômes bien utiles}
\begin{theorem}[Théorème fondamental des codes cycliques]
\pause
\underline{Unique} polynôme $g$ tel que le mot associé à $g$ engendre $C$ et $g|X^n-1$.
\vspace{0.4cm}
$g$ de degré $n-k$
\vspace{0.4cm}
$(\sigma^i(w))_{i\in \llbracket 0,k-1\rrbracket}$ base de $C$
\vspace{0.4cm}
\pause
Codage naturel associé au polynôme.
\end{theorem}
\end{frame}
\section{Réalisation informatique}
\begin{frame}
\centering\Huge Réalisation Informatique
\end{frame}
\subsection{Des structures de données}
\begin{frame}
\frametitle{Créer les structures}
\pause
Temps constant :
\begin{itemize}
\item\texttt{lor,land,lxor,lsl,lsr,=,<,>}
\item\texttt{if, for, match, \&\&}
\end{itemize}
\end{frame}
\subsubsection{Matrices et vecteurs}
\begin{frame}
\frametitle{Matrices et vecteurs}
\pause
\centering\texttt{type vecteur = int}
\centering\texttt{type matrice = int list}
\pause
\begin{itemize}
\item Calculs binaires rapides
\pause
\item Limitation par l'architecture
\pause
\item Possibilité de dépasser virtuellement.
\end{itemize}
\pause
\underline{Fonctions dévelopées: }
\pause
\begin{itemize}
\fsign{produit}{matrice \into vecteur \into vecteur}{k}
\pause
\fsign{identite}{int \into matrice}{n}
\pause
\fsign{print\_matrice}{int \into matrice \into unit}{nk}
\fsign{print\_vecteur}{int \into vecteur \into unit}{n}
\end{itemize}
\end{frame}
\subsubsection{Polynômes}
\begin{frame}
\frametitle{Polynômes}
\pause
\centering\texttt{type polynome = int}
\pause
\vspace{0.4cm}
Mêmes remarques.
\pause
\vspace{0.5cm}
\underline{Fonctions dévelopées: }
\pause
\begin{itemize}
\fsign{degre}{polynome \into int}{p}
\pause
\fsign{polmul}{polynome \into polynome \into polynome}{\min(p,q)}
\pause
\fsign{poldiveuc}{polynome \into polynome \into polynome $\times$ polynome}{p^2}
\pause
\fsign{print\_polynome}{polynome \into unit}{p}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Fonctions d'affichages}
\begin{figure}[h]
\begin{center}
\includesvg[scale=.3]{print_output_path.svg}
\caption{Exemple d'affichage de vecteur,matrice et polynômes}
\end{center}
\end{figure}
\end{frame}
\subsubsection{Codes}
\begin{frame}
\frametitle{Structures des codes}
\pause
\texttt{type code\_lineaire = \{\\\hspace{0.4cm}
k : int;\\\hspace{0.4cm}
n : int;\\\hspace{0.4cm}
g : Math.matrice;\\\hspace{0.4cm}
h : Math.matrice; \\\}
}
\pause
\vspace{0.4cm}
\texttt{type code\_cyclique = \{\\\hspace{0.4cm}
k : int;\\\hspace{0.4cm}
n : int;\\\hspace{0.4cm}
pol : Math.polynome;\\ \}
}
\pause
\vspace{0.4cm}
\begin{itemize}
\fsign{systematiqueFromRedondance}{int \into int \into matrice \into code\_lineaire}{k}
\end{itemize}
\end{frame}
\subsection{Liste des fonctions}
\begin{frame}
\frametitle{Fonctions de codage/décodage}
\pause
\begin{itemize}
\fsign{encoder}{code\_lineaire \into vecteur \into vecteur}{k}
\pause
\fsign{appartenir}{code\_lineaire -> vecteur -> bool}{n}
\pause
\fsign{distance\_minimale}{code\_lineaire -> int}{n2^{d_C}}
\pause
\fsignz{decoder}{code\_lineaire \into vecteur \into Math.vecteur}
\pause
\fsign{genererClasses}{code\_lineaire \into classes\_latérales}{n\cdot2^k}
\pause
\fsign{decoder2}{classes\_latérales \into vecteur \into vecteur}{n}
\end{itemize}
\end{frame}
\begin{frame}
\end{frame}
\begin{frame}
\frametitle{Annexes}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,language=Caml,frame=single,linerange={1-26}]{../Math.mli}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,language=Caml,frame=single,linerange={29-34}]{../Math.mli}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,language=Caml,frame=single,linerange={1-27}]{../Code.mli}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,language=Caml,frame=single,linerange={29-39}]{../Code.mli}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,linerange={48-59},language=Caml,frame=single]{../Math.ml}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,linerange={105-128},language=Caml,frame=single]{../Math.ml}
\end{frame}
\begin{frame}
\lstinputlisting[basicstyle=\scriptsize,breaklines=true,linerange={140-166},language=Caml,frame=single]{../Code.ml}
\end{frame}
\end{document}

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