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Rédaction d'une grande partie de la seconde partie du compte rendu

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Retypage de quelques fonctions
Correction d'un problème de dépendances dans le Makefile
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Mysaa 2021-06-01 20:06:47 +02:00
parent 5a16b7c99a
commit 2c825f16a8
7 changed files with 266 additions and 14 deletions
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4
Code.ml
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@ -131,8 +131,6 @@ let decoder code z =
| (_, e) -> antecedent code ((lxor) z e)
;;
let cyclencode code (x:vecteur) :vecteur = polmul code.pol x;;
end;;
@ -150,6 +148,8 @@ type code_cyclique = t;;
let get k n pol = {k=k;n=n;pol=pol};;
let cyclencode code (x:vecteur) :vecteur = polmul code.pol x;;
end;;
(*************************************************************)

2
Code.mli
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@ -8,7 +8,7 @@ type t = {
type code_lineaire = t
val encoder : code_lineaire -> Math.vecteur -> Math.vecteur
val systematiqueFromRedondance : int -> int -> int list -> code_lineaire
val systematiqueFromRedondance : int -> int -> matrice -> code_lineaire
val distance_minimale : code_lineaire -> int
val decoder : code_lineaire -> int -> Math.vecteur
val appartenir : code_lineaire -> Math.vecteur -> bool

1
CompteRendu/.gitignore vendored
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@ -5,3 +5,4 @@
*.synctex.gz
*.toc
*.thm
svg-inkscape/

71
CompteRendu/CompteRendu.tex
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@ -13,6 +13,7 @@
\usepackage{calrsfs}
\usepackage{setspace}
\usepackage{stmaryrd}
\usepackage{svg}
\usepackage{bashful}
@ -31,6 +32,8 @@
% Racourcis
\newcommand{\FD}{\mathbb{F}_2\hspace{-.2em}}
\newcommand\fsign[3]{\item \textbf{#1} : \textit{#2} \qquad Complexité en $\mathcal{O}(#3)$\\}
\newcommand\into{\textrightarrow\:}
\newtheoremstyle{break}
{\topsep}{\topsep}%
@ -80,7 +83,7 @@
$$d_C = \min_{x,y\in C\times C}\left(d(x,y)\right)$$
\end{definition}
\begin{theoreme}
\begin{theoreme}\label{thCapacité}
On peut alors exprimer $e_d$ et $e_c$ en fonction de $d_C$ :
$$ e_c = d_C - 1 \qquad e_d = \left\lfloor\frac{d_C - 1}{2}\right\rfloor$$
@ -131,7 +134,7 @@
On appelle alors \textbf{mot erreur} le mot $E = Z - Y$.
On appelle \text{syndrome} le mot $S = H \cdot Z$.
On remarque que $E$ et $Z$ on le même syndrome. On peut plus généralement définir une relation d'équivalence «avoir le même syndrome». Les classes d'équivalence de cette relation sont appelées \textbf{classes latérales}
On remarque que $E$ et $Z$ on le même syndrome. On peut plus généralement définir une relation d'équivalence «avoir le même syndrome». Les classes d'équivale nce de cette relation sont appelées \textbf{classes latérales}.
\end{definition}
\section{Codes cycliques}
@ -159,7 +162,71 @@
\part{Algorithmes}
\section{Des structures de données}
La première étape était de créer des structures de données adéquates, c'est à dire permettant d'effectuer les calculs nécessaire à l'utilisation des codes cycliques à moindre coût (temporel).
\subsection{Matrices et vecteurs}
Puisque nous avons affaire à des vecteurs de $\FD^n$ on peut les stocker sous forme d'entier «informatique». Nous sommes alors limités par la taille des mots du processeur, typiquement 32 bits ou 64 bits. Mais il est toujours possible de créer des entiers binaires virtuellement plus «grand». Un autre problème est qu'on ne peut pas récupérer la dimension d'un vecteur, et nous devons donc transmettre la donnée à coté.
Les matrices sont elles, des listes de vecteurs et sont donc naturellement stockées comme listes d'entiers binaires. On utilisera la structure de liste chainée native d'OCaml.
Nous avons ensuite codé les fonctions suivantes: (Les complexités sont données pour des matrices de $\mathcal{M}_{n,k}(\FD)$, les opérations bit à bit se faisant à temps constant, ce qui est vrai pour les entiers natifs).
\begin{itemize}
\fsign{produit}{matrice \into vecteur \into vecteur}{k} Renvoie simplement le vecteur produit $Y = M \cdot X$
\fsign{identite}{int \into matrice}{n} Renvoie la matrice identité de $\mathcal{M}_n(\FD)$
\fsign{print\_matrice}{int \into matrice \into unit}{nk} Affiche la matrice dans le terminal. Il faut spécifier la dimension verticale (des vecteurs). Un exemple de valeur de sortie est donnée figure \ref{print_matriceExemple}
\fsign{print\_vecteur}{int \into vecteur \into unit}{n} Affiche le vecteur dans le terminal, vu comme une matrice de $\mathcal{M}_{n,1}(\FD)$ (Exemple figure \ref{print_matriceExemple})
\end{itemize}
\subsection{Polynômes}
De la même manière, les polynômes sont des vecteurs de l'espace vectoriel $\FD[X]$. On peut donc eux aussi les stocker comme entiers binaires, avec l'entier écrit en base 2: $a_0a_1a_2\cdots a_d$ correspondant au polynôme $P=\displaystyle\sum^d_{i=0}a_i\cdot X^i$. Là encore, nous nous limitons aux polynômes de degré 31 ou 63, à moins d'utiliser des objets virtuels plus avancés.
Les fonctions suivantes ont été créées (les complexités sont données pour les polynômes $P$ et $Q$ de degrés respectifs $p$ et $q$).
\begin{itemize}
\fsign{degre}{polynome \into int}{p} Renvoie le degré du polynôme
\fsign{polmul}{polynome \into polynome \into polynome}{\min(p,q)} Effectue le produit de deux polynômes dans l'algèbre $\FD[X]$
\fsign{poldiveuc}{polynome \into polynome \into polynome $\times$ polynome}{p^2} Effectue la division euclidienne de $P$ par $Q$.
\fsign{print\_polynome}{polynome \into unit}{p}(Exemple figure \ref{print_matriceExemple})
\end{itemize}
\subsection{Codes}
Les différents codes (linéaires et cycliques) sont stockés comme des enregistrements.
Les codes linéaires sont la donnée de leurs matrices génératrice et une matrice de contrôle (redondante) ainsi que k et n (nécessaire car les «matrices» n'ont pas la donnée de leur hauteur). Bien que l'on puisse déduire une matrice de contrôle de la matrice génératrice, ce calcul peut se révéler coûteux.
Les codes cycliques sont simplement les données de k, de n et du polynôme générateur.
Les fonctions suivantes permettent de manipuler les structures:
\begin{itemize}
\fsign{systematiqueFromRedondance}{int \into int \into matrice \into code\_lineaire}{k} Renvoie le code linéaire systématique associé à la matrice de redondance de $\mathcal{M}_{n-k,k}$.
\end{itemize}
\section{Liste des fonctions}
Nous avons ensuite écrit des fonctions permettant de manipuler les codes linéaires:
\begin{itemize}
\fsign{encoder}{code\_lineaire \into Math.vecteur \into Math.vecteur}{}
Encode le vecteur de $\FD^k$ suivant le code linéaire spécifié, il s'agit alors d'un simple produit avec la matrice génératrice.
\fsign{appartenir}{code_lineaire -> Math.vecteur -> bool}
Renvoie \textit{vrai} si et seulement si le vecteur appartient au code, c'est à dire, si et seulement si $HX=0$ avec $H$ la matrice de contrôle et $X$ le vecteur de $\FD^n$ en question.
\fsign{distance\_minimale}{code\_lineaire -> int}{}
Renvoie la distance minimale du code, utilisée pour calculer les capacités de détéction et de correction du code (voir \ref{thCapacité})
\fsign{decoder}{code\_lineaire \into int \into Math.vecteur}{}
\end{itemize}
\part*{Annexes}
\begin{figure}[h]
\begin{center}
\label{print_matriceExemple}
\includesvg[scale=.3]{print_output.svg}
\caption{Exemple d'affichage de vecteur,matrice et polynômes}
\end{center}
\end{figure}
\end{document}

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@ -0,0 +1,177 @@
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5
Makefile
View File

@ -7,8 +7,9 @@ ascii: $(shell find textes/ -type f -iname "*.txt" ! -iname "*.ascii.txt" | sed
%.cmx: %.ml
ocamlopt -c $<
Code.cmo: Math.cmo
Code.cmo: Code.ml Math.cmo
ocamlc -c "Code.mli"
ocamlc -c "Code.ml"
%.cmi: %.mli
ocamlc -c $<
%.cmo: %.ml

16
Math.mli
View File

@ -1,20 +1,26 @@
type vecteur = int
type matrice = int list
type polynome = int
val deux_puissance : int -> int
val changer_bit : int -> int -> int
val decagauche : int -> int -> int
val nthOfBinarint : int -> int -> string
val print_matrice : int -> matrice -> unit
val print_vecteur : int -> vecteur -> unit
val print_polynome : polynome -> unit
val produit : matrice -> vecteur -> vecteur
val deux_puissance : int -> int
val identite : int -> matrice
val orderize : 'a -> 'a -> 'a * 'a
val identite : int -> int list
val changer_bit : int -> int -> int
val decagauche : int -> int -> int
val respecter : int -> int list -> bool
val matriceAuPif : int -> int -> matrice
val polmul : polynome -> polynome -> polynome
val degre : polynome -> int
val polmul : polynome -> polynome -> polynome
val poldiveuc : polynome -> polynome -> polynome * polynome
val poldiv : polynome -> polynome -> polynome
val polrst : polynome -> polynome -> polynome