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.
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