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"Maths.ml";; (* La bonne structure *) -type code_lineaire = {k : int; n : int; g : matrice; h : matrice};; +type code_lineaire = {klin : int; nlin : int; g : matrice; h : matrice};; + +(* La super stucture *) +type code_cyclique = {kcyc : int; ncyc: int; pol: polynome};; (* Calcule Y = GX *) let encoder code x = produit code.g x ;; @@ -44,9 +47,34 @@ let construire_code_lineaire_systematique k n redondance = c :: (iteredon (ajout * 2) queue) in iteredon 1 redondance and h = redondance @ (identite (n-k)) in - {k = k; n = n; g = g; h = h} + {klin = k; nlin = n; g = g; h = h} ;; +(* Construit le code linéaire associé au code cyclique *) +let cycliqueVersLineaire code = + let k=code.kcyc and n=code.ncyc and pol=code.pol in + let g = + let rec itdecal i l = + if i<0 + then l + else itdecal (i-1) ((pol lsl i)::l) + in itdecal (k-1) [] + and h = + let polh = poldiv ((deux_puissance n) + 1) pol in + print_polynome polh; + let filtre = (deux_puissance (n-k+1))-1 in + let rec sub i l = + if i<(-k) + then l + else sub (i-1) (((decagauche polh i) land filtre)::l) + in sub (n-k-1) [] + in {klin = k; nlin = n; g = g; h = h} +;; + + +let lineaireVersCyclique code = + (*TODO*)1 +;; (* Etant donnés tous les vecteurs de poids p dans un espace de dimension d, retourne touts ceux de poids p+1 dans ce même espace *) @@ -88,7 +116,7 @@ let appartenir code v = produit code.h v = 0;; (* Calcul la distance minimale d'un code *) let distance_minimale code = - let n = code.n in + let n = code.nlin in let propriete = fun v -> (0 < v) && (appartenir code v) in let (p, _) = plus_petit_verifiant propriete n n in p ;; @@ -97,7 +125,7 @@ exception PasDansLeCodeException;; (* Calcul de façons à ouf l'antécédent de 'y' pour le code 'code' *) let antecedent code y = - let mot_max = (deux_puissance code.k) - 1 in + let mot_max = (deux_puissance code.klin) - 1 in let rec iterer = function | x when x = mot_max -> raise PasDansLeCodeException | x -> @@ -111,12 +139,12 @@ exception IndecodableException;; (* Applique notre algorithme préféré *) let decoder code z = let d_min = distance_minimale code in - let e_c = (d_min - 1) / 2 and n = code.n in + let e_c = (d_min - 1) / 2 and n = code.nlin in let propriete = fun v -> appartenir code ((lxor) z v) in match plus_petit_verifiant propriete e_c n with | (-1, 0) -> raise IndecodableException | (_, e) -> antecedent code ((lxor) z e) ;; - + (* Voilà *) \ No newline at end of file diff --git a/CompteRendu/.gitignore b/CompteRendu/.gitignore new file mode 100644 index 0000000..90ee7eb --- /dev/null +++ b/CompteRendu/.gitignore @@ -0,0 +1,6 @@ +*.aux +*.log +*.pdf +*.sh +*.synctex.gz +*.toc diff --git a/CompteRendu/CompteRendu.tex b/CompteRendu/CompteRendu.tex new file mode 100644 index 0000000..14f50af --- /dev/null +++ b/CompteRendu/CompteRendu.tex @@ -0,0 +1,49 @@ +% !TeX encoding = UTF-8 +\documentclass[10pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage[margin=0.5in]{geometry} +\usepackage[french]{babel} +\usepackage{tabularx} +\usepackage{theorem} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{calrsfs} +\usepackage{setspace} + +\usepackage{bashful} + + +\theorembodyfont{\upshape} + +% Assure que les nouvelles parties commencent toujours sur une nouvelle page. +\let\oldpart\part +\renewcommand\part{\clearpage\oldpart} + +% Un peu plus d'interligne, c'est plus lisible +\doublespacing + +% Fait une jolie barre horizontale +\newcommand{\hsep}{\centerline{\rule{0.8\linewidth}{.05pt}}} + +\author{Samy AVRILLON, Victor BELLOT, Dylan THEVENET} +\title{Études des codes cycliques} +\begin{document} + + \maketitle + + \hsep + + \tableofcontents + + \pagebreak + + \part{Mathématique} + + + + \part{Algorithmie} + \section{Structures de données crées} + \section{Liste des fonctions} + + +\end{document} diff --git a/Maths.ml b/Maths.ml index e122d67..7a3816f 100644 --- a/Maths.ml +++ b/Maths.ml @@ -1,6 +1,9 @@ type vecteur = int;; type matrice = int list;; +type polynome = int;; + +(***************** AFFICHAGE *****************) (* Zolie fonction d'affichage (h=nombre de lignes) *) (* L'algorithme est moche, mais y a pas trop d'autre solution que de arrayiser la liste, c'est stoqué dans le mauvais sens *) let nthOfBinarint i n = if ((0b1 lsl n) land i)=0 then "0" else "1";; @@ -26,6 +29,22 @@ let print_matrice h (matl:matrice) :unit = print_endline "┘";; let print_vecteur h x = print_matrice h [x];; +let print_polynome pol = + let rec aux pol i preums = match pol with + | 0 -> print_endline "" + | pol when pol mod 2 = 0 -> aux (pol lsr 1) (i+1) preums + | _ -> begin + begin + match (preums,i) with + | true,0 -> print_string "1" + | true,_ -> (print_string "X^";print_int i) + | false,_ -> (print_string " + X^";print_int i) + end;aux (pol lsr 1) (i+1) false + end + in aux pol 0 true;; + +(***************** MATRICES *****************) + (* Effectue le produit matriciel 'matrice' . 'vecteur' *) let produit (matrice:matrice) (vecteur:vecteur) :vecteur = let rec auxiliaire resultat_partiel masque = function @@ -35,13 +54,17 @@ let produit (matrice:matrice) (vecteur:vecteur) :vecteur = if masque mod 2 = 1 then (lxor) colonne resultat_partiel else resultat_partiel - in auxiliaire resultat (masque / 2) reste + in auxiliaire resultat (masque lsr 1) reste in auxiliaire 0 vecteur matrice ;; (* Ne calcul papy *) let deux_puissance = (lsl) 1;; +let orderize p q = + if (p x lsr (-n) + | n when n > 0 -> x lsl n + | n -> x +;; (* Vérifie que 'y' respecte toutes les contraintes de 'cs' *) -(* Est-ce que pour tout P dans cs, P(cs) ? *) +(* Est-ce que pour tout P dans cs, P(y) ? *) let respecter y cs = let ny = (lnot) y in List.fold_right (fun c b -> b && (land) c ny > 0) cs true @@ -74,3 +99,46 @@ let matriceAuPif l h = | 0 -> tmp | l -> aux (l-1) h (((Random.bits ()) land filtre)::tmp) in aux l h [];; + +(***************** POLYNOMES *****************) + +let polmul (p_in:polynome) (q_in:polynome) : polynome = + let (p,q) = orderize p_in q_in in + let rec sub pa qa somme = + if pa=0 then somme else + sub (pa lsr 1) (qa lsl 1) (if (pa mod 2=0) then somme else (somme lxor qa)) + in sub p q 0 ;; + +let degre (p:polynome) :int = + let rec aux p d = + match p with + | 0 -> -1 + | 1 -> d + | _ -> aux (p lsr 1) (d+1) + in aux p 0;; + +let poldiveuc (p:polynome) (q:polynome) : (polynome * polynome) = + let dq = degre q in + let rec sub quotient reste = + let dr = degre reste in + let d = dr - dq in + if d >= 0 + then sub (quotient lxor (deux_puissance d)) (reste lxor (q lsl d)) + else (quotient, reste) + in sub 0 p;; + +let poldiv (p:polynome) (q:polynome) : polynome = fst (poldiveuc p q);; +let polrst (p:polynome) (q:polynome) : polynome = snd (poldiveuc p q);; + + + + + + + + + + + + + diff --git a/Test.ml b/Test.ml index ebc6c80..69e421d 100644 --- a/Test.ml +++ b/Test.ml @@ -6,6 +6,15 @@ let matest = [0b01110; 0b00101; 0b10111];; print_matrice 5 matest;; produit matest 0b110;; (* -> 0b10010 = 8*) +(* Tests des polynomes *) +let pol1 = 13 and pol2 = 67;; +print_polynome pol1;; +print_polynome pol2;; +print_polynome (polmul pol1 pol2);; +let qt,rst=(poldiveuc pol2 pol1) in + print_polynome qt; + print_polynome rst;; + (* Test des fonctions de base *) deux_puissance 11;; identite 3;; @@ -29,11 +38,13 @@ let code_hamming = construire_code_lineaire_systematique 4 7 [7; 3; 5; 6] ;; -distance_minimale code_paparfait;; +distance_minimale code_hamming;; + + exception GTROUVE of matrice;; -for i=0 to 200000 +for i=0 to 20 do Random.self_init (); let matPapa = matriceAuPif 9 12 in @@ -48,10 +59,19 @@ print_vecteur 21 (encoder code_paparfait 0b011011011001);; print_matrice 3 (suivants 3 (suivants 3 (suivants 3 (suivants 3 [0b000]))));; - - print_vecteur 7 (encoder code_hamming 0b0100);; decoder code_hamming 0b1010100;; decoder code_hamming 0b0010100;; decoder code_hamming 0b1110000;; print_vecteur 7 21;; + +(* Tests des codes cycliques *) +let cocycl = {ncyc=7;kcyc=4;pol=13};; +print_polynome cocycl.pol;; +poldiveuc ((deux_puissance 7) +1) cocycl.pol;; +print_polynome ((deux_puissance 7) +1);; +print_polynome (poldiv ((deux_puissance 7) +1) cocycl.pol);; +let cocylined = cycliqueVersLineaire cocycl;; +print_matrice 7 cocylined.g;; +print_matrice 4 cocylined.h;; +distance_minimale cocylined;;