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Nouvel algorithme de décodage avec enregistrement dans un dictionnaire des syndromes

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Fin de la rédaction du dossier (manque encore des finitions)
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Mysaa 2021-06-02 15:12:22 +02:00
parent 2c825f16a8
commit 36875fc40a
9 changed files with 307 additions and 145 deletions
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35
Code.ml
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@ -22,8 +22,8 @@
* on représente un vecteur dans F_2 par un entier dont la décomposition binaire correspond aux composantes du vecteur * on représente un vecteur dans F_2 par un entier dont la décomposition binaire correspond aux composantes du vecteur
* on représente une matrice par un liste d'entier, il s'agit de la liste de ses colonnes (qui sont donc des vecteurs) * on représente une matrice par un liste d'entier, il s'agit de la liste de ses colonnes (qui sont donc des vecteurs)
*)
(*
#cd "/home/mysaa/Documents/Arbeiten/TIPE2021/";; #cd "/home/mysaa/Documents/Arbeiten/TIPE2021/";;
#load "Math.cmo";; #load "Math.cmo";;
*) *)
@ -132,6 +132,37 @@ let decoder code z =
;; ;;
(*** Décodage avec stockage des syndromes ***)
(* syndr est une fonction qui à un vecteur associe son syndrome.
En effet, la notion de syndrome dépend de quelle matrice H nous avons choisi.
Créer cette fonction ici permet de ne plus dépendre de la matrice H de code_lineaire *)
type classesLaterales = {syndr : Math.vecteur -> Math.vecteur;
erreur : Math.vecteur -> Math.vecteur;
babr : Math.vecteur Math.binabr};;
let genererClasses code =
let syndr = function z -> Math.produit code.h z in
let compacteur babr v = if BFeuille<>babr && not (appartenir code v)
then (print_int v;putWho babr (syndr v)) v false
else BFeuille in
let rec iter babr0 lst =
let svts = suivants code.n lst in
let babr = List.fold_left compacteur babr0 svts in
if babr<>BFeuille
then iter babr svts
else babr0 (*On a dépassé la capacité de correction*)
in
let babr = iter (put BFeuille 0 0) [0] in
let erreur = function z -> get babr z
in {syndr=syndr;erreur=erreur;babr=babr}
;;
let decoder2 classes z :vecteur=
let s = classes.syndr z in
let e = classes.erreur s in
z lxor e;;
end;; end;;
(*************************************************************) (*************************************************************)

14
Code.mli
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@ -6,12 +6,22 @@ type t = {
g : Math.matrice; g : Math.matrice;
h : Math.matrice; } h : Math.matrice; }
type code_lineaire = t type code_lineaire = t
exception PasDansLeCodeException
val encoder : code_lineaire -> Math.vecteur -> Math.vecteur val encoder : code_lineaire -> Math.vecteur -> Math.vecteur
val systematiqueFromRedondance : int -> int -> matrice -> code_lineaire val systematiqueFromRedondance : int -> int -> Math.matrice -> code_lineaire
val distance_minimale : code_lineaire -> int val distance_minimale : code_lineaire -> int
val decoder : code_lineaire -> int -> Math.vecteur val decoder : code_lineaire -> Math.vecteur -> Math.vecteur
val appartenir : code_lineaire -> Math.vecteur -> bool val appartenir : code_lineaire -> Math.vecteur -> bool
type classesLaterales = {
syndr : Math.vecteur -> Math.vecteur;
erreur : Math.vecteur -> Math.vecteur;
babr : Math.vecteur Math.binabr;
}
val genererClasses : t -> classesLaterales
val decoder2 : classesLaterales -> Math.vecteur -> Math.vecteur
end end
module CCyclique : module CCyclique :

1
CompteRendu/.gitignore vendored
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@ -6,3 +6,4 @@
*.toc *.toc
*.thm *.thm
svg-inkscape/ svg-inkscape/
print_output_path.svg

25
CompteRendu/CompteRendu.tex
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@ -123,6 +123,7 @@
$$d_C \leqslant n+1-k$$ $$d_C \leqslant n+1-k$$
Enfin, on peut utiliser que la distance minimale d'un code linéaire est le nombre minimal de colonnes linéairement dépendantes de la matrice de contrôle $H$.
\end{theoreme} \end{theoreme}
Voici ici quelques concepts qui seront utiles au décodage. Voici ici quelques concepts qui seront utiles au décodage.
@ -149,11 +150,10 @@
On définit le mot binaire associé au polynôme $P=\displaystyle\sum_{i=0}^{n-1}{a_i\cdotp X^i}$ comme étant le mot $a_0a_1\cdots a_{n-1}$. Le polynôme réciproquement associé à un mot binaire par ce procédé est appelé \textbf{représentation polynomiale} du mot. On définit le mot binaire associé au polynôme $P=\displaystyle\sum_{i=0}^{n-1}{a_i\cdotp X^i}$ comme étant le mot $a_0a_1\cdots a_{n-1}$. Le polynôme réciproquement associé à un mot binaire par ce procédé est appelé \textbf{représentation polynomiale} du mot.
\end{definition} \end{definition}
\begin{definition}[Polynôme générateur]
Un polynôme $P$ de $\FD\,[X]$ de degré $n-k$ est dit \textbf{générateur du code cyclique $C$ de paramètre $(k,n)$} lorsque $P | X^n + 1$ et que $(\sigma^i(w))_{i\in \llbracket 0,k-1\rrbracket}$ est une base de $C$ avec $\sigma$ l'opérateur de décalage binaire cyclique est $w$ le mot associé à $P$
\end{definition}
\begin{theoreme}[Théorème fondamental des codes cycliques] \begin{theoreme}[Théorème fondamental des codes cycliques]
Tout code cyclique admet un et un seul polynôme générateur. Pour tout code cyclique $C$ de paramètres $(n,k)$, il existe un unique polynôme $g$ tel que le mot associé à $g$ engendre $C$ et que $g|X^n-1$. Ce polynôme est alors appelé \textbf{polynôme générateur du code cyclique $C$}.
On démontre que ce $g$ est alors de degré $n-k$ et que $(\sigma^i(w))_{i\in \llbracket 0,k-1\rrbracket}$ est une base de $C$ avec $\sigma$ l'opérateur de décalage binaire cyclique est $w$ le mot associé à $y$
\end{theoreme} \end{theoreme}
\begin{remarque} \begin{remarque}
@ -211,20 +211,25 @@
Nous avons ensuite écrit des fonctions permettant de manipuler les codes linéaires: Nous avons ensuite écrit des fonctions permettant de manipuler les codes linéaires:
\begin{itemize} \begin{itemize}
\fsign{encoder}{code\_lineaire \into Math.vecteur \into Math.vecteur}{} \fsign{encoder}{code\_lineaire \into vecteur \into vecteur}{k}
Encode le vecteur de $\FD^k$ suivant le code linéaire spécifié, il s'agit alors d'un simple produit avec la matrice génératrice. 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} \fsign{appartenir}{code\_lineaire -> vecteur -> bool}{n}
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. 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}{} \fsign{distance\_minimale}{code\_lineaire -> int}{non}
Renvoie la distance minimale du code, utilisée pour calculer les capacités de détéction et de correction du code (voir \ref{thCapacité}) Renvoie la distance minimale du code, utilisée pour calculer les capacités de détéction et de correction du code (voir \ref{thCapacité}). Le calcul s'effectue en recherchant le plus petit mot (au sens du poids) qui soit dans le code.
\fsign{decoder}{code\_lineaire \into int \into Math.vecteur}{} \fsign{decoder}{code\_lineaire \into vecteur \into Math.vecteur}{euhhhhhhhh}
Décode le mot de $\FD^n$ donné. L'algorithme recherche le mot le plus proche du mot recu qui soit dans le code et à une distance inferieure à la distance de correction. Si il est impossible de décoder, renvoie une exception du type \verb|IndecodableException|.
\fsign{genererClasses}{code\_cyclique \into classes\_latérales}{well ...}Génère les classes latérales associées au code spécifié. Parcours les vecteurs par poids croissant (d'abord ceux de poids 1, puis 2, etc…) et référence leur syndrome dans \verb|classes.rpz : vecteur -> vecteur| qui à un syndrome associe le représentant de la classe latérale de poids le plus faible.
\fsign{decoder2}{classes\_latérales \into vecteur \into vecteur}{n\cdot \ln(n)} Décode le mot $Z$ de $\FD^n$ donné. L'algorithme calcule le syndrome du mot, obtient le représentant de la classe latérale associée à ce vecteur $E$. Dans l'hypothèse où le mot reçu est effectivement décodable, alors, son «erreur» $E = Z-Y$, avec $Y$ le mot du code initialement envoyé, a la même classe latérale que $Z$. Donc $Y = Z+E$, on obtient un mot du code que l'on peut décoder.
\end{itemize} \end{itemize}
\part*{Annexes} \part*{Annexes}
\begin{figure}[h] \begin{figure}[h]
\begin{center} \begin{center}
\label{print_matriceExemple} \label{print_matriceExemple}
\includesvg[scale=.3]{print_output.svg} \includesvg[scale=.3]{print_output_path.svg}
\caption{Exemple d'affichage de vecteur,matrice et polynômes} \caption{Exemple d'affichage de vecteur,matrice et polynômes}
\end{center} \end{center}
\end{figure} \end{figure}

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CompteRendu/print_output.svg
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10
Makefile
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@ -7,14 +7,12 @@ ascii: $(shell find textes/ -type f -iname "*.txt" ! -iname "*.ascii.txt" | sed
%.cmx: %.ml %.cmx: %.ml
ocamlopt -c $< ocamlopt -c $<
Code.cmo: Code.ml Math.cmo Code.cmo: Code.ml Code.mli Math.cmo
ocamlc -c "Code.mli" ocamlc -c "Code.mli"
ocamlc -c "Code.ml" ocamlc -c "Code.ml"
%.cmi: %.mli Math.cmo: Math.ml Math.mli
ocamlc -c $< ocamlc -c Math.mli
%.cmo: %.ml ocamlc -c Math.ml
if [ -f $*".mli" ]; then ocamlc -c $*".mli" ; fi
ocamlc -c $<
tipe: tipe.cmx tipe: tipe.cmx
ocamlopt -o tipe str.cmxa tipe.cmx ocamlopt -o tipe str.cmxa tipe.cmx

43
Math.ml
View File

@ -130,11 +130,44 @@ let poldiveuc (p:polynome) (q:polynome) : (polynome * polynome) =
let poldiv (p:polynome) (q:polynome) : polynome = fst (poldiveuc p q);; let poldiv (p:polynome) (q:polynome) : polynome = fst (poldiveuc p q);;
let polrst (p:polynome) (q:polynome) : polynome = snd (poldiveuc p q);; let polrst (p:polynome) (q:polynome) : polynome = snd (poldiveuc p q);;
(***************** Arbres de recherche de 'vecteurs' *****************)
type 'a binabr = BNoeud of 'a binabr * 'a binabr | BVal of 'a * 'a binabr | BFeuille;;
let isEmpty babr = babr=BFeuille;;(* Ne renvoie que les biens fondés, c'est à dire sans Noeud vide*)
exception NoSuchKeyException;;
let rec get ba (k:vecteur) =
match (ba,k) with
| (BNoeud(_,_) ,0)
| (BFeuille ,_) -> raise NoSuchKeyException
| (BVal(v,rr) ,0) -> v
| (BVal(v,rr) ,_) -> get rr k
| (BNoeud(t0,t1),_) ->
match k mod 2 with
| 0 -> get t0 (k lsr 1)
| 1 -> get t1 (k lsr 1)
| _ -> failwith "Félicitations, vous avez cassé les maths";;
let rec putWho ba (k:vecteur) v keepNew =
match (ba,k) with
| (BVal(old,rr) ,0) -> if keepNew then BVal(v,rr) else BVal(old,rr)
| (BFeuille ,0)
| (BNoeud(_,_) ,0) -> BVal(v,ba)
| (BVal(o,rr) ,_) -> BVal(o,putWho rr k v keepNew)
| (BFeuille ,_) -> putWho (BNoeud(BFeuille,BFeuille)) k v keepNew
(*match k mod 2 with
| 0 -> BNoeud(putWho BFeuille (k lsr 1) v keepNew, BFeuille)
| 1 -> BNoeud(BFeuille, putWho BFeuille (k lsr 1) v keepNew)
| _ -> failwith "Ich gratuliere Sie. Sie haben Mathe zerbrochen"
end*)
| (BNoeud(t0,t1),_) ->
match k mod 2 with
| 0 -> BNoeud(putWho t0 (k lsr 1) v keepNew, t1)
| 1 -> BNoeud(t0, putWho t1 (k lsr 1) v keepNew)
| _ -> failwith "99 7'4 c0mp1373m3n7 c4553 135 m47h5. 7u 73 23nd5 c0mp73 ?!";;
let put ba k v = putWho ba k v true;; (* Par défaut, on garde le nouveau *)

8
Math.mli
View File

@ -24,3 +24,11 @@ val polmul : polynome -> polynome -> polynome
val poldiveuc : polynome -> polynome -> polynome * polynome val poldiveuc : polynome -> polynome -> polynome * polynome
val poldiv : polynome -> polynome -> polynome val poldiv : polynome -> polynome -> polynome
val polrst : polynome -> polynome -> polynome val polrst : polynome -> polynome -> polynome
type 'a binabr = BNoeud of 'a binabr * 'a binabr | BVal of 'a * 'a binabr | BFeuille
exception NoSuchKeyException
val get : 'a binabr -> vecteur -> 'a
val putWho : 'a binabr -> vecteur -> 'a -> bool -> 'a binabr
val put : 'a binabr -> vecteur -> 'a -> 'a binabr

22
Test.ml
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@ -6,6 +6,24 @@ Sys.command "make Math.cmo Code.cmo";;
open Math;; open Math;;
open Code;; open Code;;
let print_boolean b = match b with
| true -> print_string "true"
| false -> print_string "false";;
let codeh = CLineaire.systematiqueFromRedondance 4 7 [7; 3; 5; 6];;
let vecteurs = matriceAuPif 9 (8-1);;
print_matrice 4 vecteurs;;
let classes = CLineaire.genererClasses codeh;;
let test v = try
CLineaire.encoder codeh (CLineaire.decoder codeh v) <> CLineaire.decoder2 classes v
with
Code.CLineaire.PasDansLeCodeException -> (print_int v;print_endline "*\n";false) in
List.find_opt test vecteurs;;
CLineaire.decoder codeh 119;;
CLineaire.encoder codeh 1;;
CLineaire.decoder2 classes 119;;
exit 0;;
(* Test du produit de matrice *) (* Test du produit de matrice *)
let matest = [0b01110; 0b00101; 0b10111];; let matest = [0b01110; 0b00101; 0b10111];;
print_matrice 5 matest;; print_matrice 5 matest;;
@ -16,7 +34,7 @@ let pol1 = 13 and pol2 = 67;;
print_polynome pol1;; print_polynome pol1;;
print_polynome pol2;; print_polynome pol2;;
print_polynome (polmul pol1 pol2);; print_polynome (polmul pol1 pol2);;
let qt,rst=(poldiveuc pol2 pol1) in let qt,rst=(poldiveuc pol2 pol1) in
print_polynome qt; print_polynome qt;
print_polynome rst;; print_polynome rst;;
@ -78,7 +96,7 @@ print_matrice 4 cocylined.h;;
CLineaire.distance_minimale cocylined;; CLineaire.distance_minimale cocylined;;
print_vecteur 7 (cyclencode cocycl 0b1010);; print_vecteur 9 203;;
(* Essayons de générer une table d'addition *) (* Essayons de générer une table d'addition *)