2
PASCALE CHARPIN
zeros of the products
xy,
for a fixed
x.
Through Formulae (I) and (II), we
will establish some relations between the possible values of
xy
and the set of
codewords orthogonal to x.
The formulae are most interesting for codes
C
which contain the RM-code of
order
j
and are contained in the RM-code of order
j
+ 1, for some
j.
In this
case the code C is an ideal of A. Moreover we often will suppose that C is an
extended cyclic code; then we can use together the properties of RM-codes and
the permutations which conserve the code C.
The principal results are obtained in the binary case, when C.L is a subcode
of the Reed-Muller code of order 2. For instance we were able to verify the con-
jecture of CAMION, COURTEAU and MONTPETIT [6]: form even there are eight
distinct weight distributions for the cosets of any 2-error-correcting extended
BCH codes of length 2m [10]. We give here general tools we applied to the
special case of 2-error-correcting extended BCH codes; we explain the possible
applications and give some new results. Some examples with numerical results
are given in
[12].
2. Terminology and basic properties
2.1. Extended cyclic codes in a group algebra. We denote by
A
the
group algebra
K[G,
+],which is the set of formal polynomials
with
and
(1)
X
=
L
XgX
9
, Xg
E
K,
gEG
0
=
L
OX9
,
1
=
L
X
9
,
aX9
+
bX9
=
(a
+
b )X9

gEG gEG
A
code of A is a K -subspace of A. Let C be such a code; we will denote by
c.L
the dual of C:
c.L
=
{yEA
I
x,
y
=
0, for all
X
E
c}
where x, y
=
LgEG
x9 y9
.
The algebra A has only one maximal ideal, its so-called mdical
(2)
P
= {
x
E
A
I
L
x
9
=
0 }
= {
x
E
A
I
xP
=
0 } .
gEG
For every
j,
we denote by
p1
the jth power of
P;
it is the ideal of
A
generated
by the products
TI{=
1
xi, xi
E
P.
One obtains the decreasing sequence of ideals
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