6 PASCALE CHARPIN

3. Weight polynomials of cosets

3.1. Translations on codewords. Let h be any non zero element of G. We

denote by Th the permutation on G which acts as follows on the elements of A:

We will say that Th is an h-translation on the codewords. An ideal of

A

is a code

invariant under every Th.

Let x and y in

A.

It follows easily from the definition of the multiplication in

A

that

xy

=

L (LXgYh-g) xh

=

L x,Xhy xh

=

L x,rh(Y) xh.

hEG gEG hEG hEG

Therefore we obtain a relation which links up the weight of xy with the set of

codewords orthogonal to

x:

(I)

w(xy) =card { g

E

G

I

x, rh(Y)

-=1- 0 } .

Suppose that the code C is an ideal of A. Then the dual of C is also an ideal. Let

y E Cj_; so the set of the rh(Y) is a subset of codewords of Cj_ of same weight.

So Formula (I) means that the weight of xy equals the number of elements of

that subset which are orthogonal to x. In the applications we study later, we

always consider the following ideals C of A:

PROPOSITION

1. Let y in

A

such that depth(y)

= j

with

j

E

[0,

m(p

-1) -1].

Then

Th(y+Pj+

1

)

=

y+Pj+

1

'

\:/hE G.

Therefore let C be a code of A of depth

j

satisfying

pJ+

1

c

c

c

pj ( i.e.

pm(p-

1)-1+ 1

c

cl_

c

pm(p-

1)-j) .

Then C is an ideal of A, so that Cl_ is also an ideal.

Proof: Let hE G. Note that Xh- 1 is an element of P. Then

Xhy

=

(Xh- 1)y

+

y where (Xh- 1)y E

pJ+

1

.

Thus the coset y

+

P1+

1

is invariant under any h-translation. The code C can

be considered as a union of such cosets. Hence it is an ideal of

A.

0

Remark: Assume that p

=

2. It follows immediatly from the proposition

above that any coset of depth

j

of the RM-code P1+

1

is an orphan (see the

terminology in [5]). Indeed the minimum weight codewords of these cosets cover

all coordinate positions.

3. Weight polynomials of cosets

3.1. Translations on codewords. Let h be any non zero element of G. We

denote by Th the permutation on G which acts as follows on the elements of A:

We will say that Th is an h-translation on the codewords. An ideal of

A

is a code

invariant under every Th.

Let x and y in

A.

It follows easily from the definition of the multiplication in

A

that

xy

=

L (LXgYh-g) xh

=

L x,Xhy xh

=

L x,rh(Y) xh.

hEG gEG hEG hEG

Therefore we obtain a relation which links up the weight of xy with the set of

codewords orthogonal to

x:

(I)

w(xy) =card { g

E

G

I

x, rh(Y)

-=1- 0 } .

Suppose that the code C is an ideal of A. Then the dual of C is also an ideal. Let

y E Cj_; so the set of the rh(Y) is a subset of codewords of Cj_ of same weight.

So Formula (I) means that the weight of xy equals the number of elements of

that subset which are orthogonal to x. In the applications we study later, we

always consider the following ideals C of A:

PROPOSITION

1. Let y in

A

such that depth(y)

= j

with

j

E

[0,

m(p

-1) -1].

Then

Th(y+Pj+

1

)

=

y+Pj+

1

'

\:/hE G.

Therefore let C be a code of A of depth

j

satisfying

pJ+

1

c

c

c

pj ( i.e.

pm(p-

1)-1+ 1

c

cl_

c

pm(p-

1)-j) .

Then C is an ideal of A, so that Cl_ is also an ideal.

Proof: Let hE G. Note that Xh- 1 is an element of P. Then

Xhy

=

(Xh- 1)y

+

y where (Xh- 1)y E

pJ+

1

.

Thus the coset y

+

P1+

1

is invariant under any h-translation. The code C can

be considered as a union of such cosets. Hence it is an ideal of

A.

0

Remark: Assume that p

=

2. It follows immediatly from the proposition

above that any coset of depth

j

of the RM-code P1+

1

is an orphan (see the

terminology in [5]). Indeed the minimum weight codewords of these cosets cover

all coordinate positions.