TP 4

restart;

Mise en place du corps K

p:=3;m:=5;

IiIk

IiIm

P:=X^5+X^4+X^2+1;

LCoqJClJIlhHNiIiIiYiIiJGKCokKUYlIiIlRihGKCokKUYlIiIjRihGKEYoRig=

alias(alpha=RootOf(P,X) mod p);#d\303\251finit l'extension K=F_p[alpha] de degr\303\251 m

SSZhbHBoYUc2Ig==

p^m;

IiRWIw==

Power(alpha,p^m-1) mod p ;Power(alpha,(p^m-1)/2) mod p;

IiIi

IiIj

n:=p^m-1;

IiRVIw==

T:=Array(0..n-1,i->Power(alpha,i) mod p):

LD:=proc(x)

global T,n;

local i;

for i from 0 to n-1 do

if T[i]=x then return i end if;

end do;

FAIL

end proc:

F:=i->p*i mod n;

Zio2I0kiaUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkbW9kR0YlNiQqJkkicEdGJSIiIkYkRi5JIm5HRiVGJUYlRiU=

Orb:=proc(i::nonnegint)

description "calcule les classes cyclotomipues";

global n;local Omega,j;

if i>=n then error "mauvais argument" end if;

Omega:=[];j:=i;

do

Omega:=[op(Omega),j];

j:=F(j);

if member(j,Omega) then return Omega end if;

end do;

FAIL;

end proc:

Orb(6);

NyciIiciIz0iI2EiJGkiIiIj

Construction du code BCH

la distance apparente du code BCH

delta:=8;

IiIp

L:=[seq(Orb(i),i=1..delta-1)];

Nyk3JyIiIiIiJCIiKiIjRiIjIik3JyIiIyIiJyIjPSIjYSIkaSI3J0YlRiZGJ0YoRiQ3JyIiJSIjNyIjTyIkMyIiIyMpNyciIiYiIzoiI1giJE4iIiRqIjcnRitGLEYtRi5GKjcnIiIoIiNAIiNqIiQqPSIjJCk=

J :=convert({op(map(op,L))},list);

NzsiIiIiIiMiIiQiIiUiIiYiIiciIigiIioiIzciIzoiIz0iI0AiI0YiI08iI1giI2EiI2oiIyIpIiMjKSIjJCkiJDMiIiROIiIkaSIiJGoiIiQqPQ==

JSFH

les racines du code

rac:=map(i -> T[i],J);

NztJJmFscGhhRzYiKiQpRiMiIiMiIiIqJClGIyIiJEYoKiQpRiMiIiVGKCwoKiZGJ0YoRi1GKEYoKiZGJ0YoRiZGKEYoRidGKCwsRixGKComRidGKEYqRihGKEYlRigqJkYnRihGI0YoRihGKEYoLCxGLEYoRilGKEYlRihGI0YoRidGKCwoRixGKEYxRihGNEYoLCpGLEYoRilGKEYxRihGI0YoLCZGLEYoRihGKCwsRixGKEYzRihGJUYoRiNGKEYnRigsKEYpRihGMUYoRjRGKCwqRilGKEYlRihGNEYoRidGKCwoRixGKEYjRihGJ0YoLCxGLEYoRilGKEYxRihGI0YoRihGKCwqRixGKEYpRihGJUYoRiNGKCwmRiVGKEYnRigsKEYwRihGKUYoRiNGKCwoRjBGKEYxRihGKEYoLCxGLEYoRjNGKEYlRihGI0YoRihGKCwsRixGKEYzRihGMUYoRiNGKEYnRigsJkYsRihGI0YoLCpGKUYoRjFGKEY0RihGKEYoLCpGLEYoRjNGKEYxRihGI0YoLCpGLEYoRjNGKEYlRihGNEYo

le polyn\303\264me g\303\251n\303\251rateur

g:=sort(Expand(mul(X-r,r in rac)) mod p,X);

LEYqJClJIlhHNiIiI0QiIiJGKCokKUYlIiNDRihGKCokKUYlIiNBRihGKCokKUYlIiNARihGKCokKUYlIiM/RihGKComIiIjRigpRiUiIz1GKEYoKiZGNkYoKUYlIiM8RihGKComRjZGKClGJSIjO0YoRigqJkY2RigpRiUiIzpGKEYoKiQpRiUiIzhGKEYoKiZGNkYoKUYlIiM3RihGKCokKUYlIiM1RihGKComRjZGKClGJSIiKUYoRigqJClGJSIiJUYoRigqJClGJSIiJEYoRigqJClGJUY2RihGKComRjZGKEYlRihGKEYoRig=

la dimension

k:=n-degree(g,X);

IiQ8Iw==

le nombre de mots du code

p^k;

ImNxakBKTCNbIz4ibzE6QzM9cXF1bl5xOVdYZk1yWkByUzdRMTMlRzJjIVxAaSgqKnpXbDUmPVFWLFYk