4
PASCALE CHARPIN
[0,
n]:
for any s
E
S, let I.::,~ 1 sipi , Si
E
[O,p-
1] , be the p-ary expansion of
s;
then the p-weight of
s
is
m-1
wp(s) =Lsi.
i=O
DEFINITION
3. The p-ary Reed-Muller code of order r, denoted by Rp(r, m),
is the extended cyclic code in A, whose defining-set is
lp(j, m)
= {
s
E
S
I
wp(s)
j } , j
=
m(p- 1)- r .
BERMAN
proved in
[4]
that the p-ary Reed-Muller codes are the powers of the
radical of
A.
More precisely:
(3)
pi
=
Rp(m(p- 1) -
j,
m) , for all
j
E
[1, m(p- 1)]
Therefore, we can deduce from a wellknown property of the generalized Reed-
Muller codes that (Pi)j_
=
pm(p- 1)-i+ 1
.
In the following we will identify the
p-ary RM-codes with the codes pi. Note that
(4)
dim pi =card { s
E
S
I
wp(s)
?_
j }
Cosets of extended cyclic codes and equations over finite fields: Let
C
be an extended cyclic code with defining set
T.
Let x
E
A
and consider the
coset
Cx
=
x
+
C .
We have
¢s(z)
=
¢s(x) ,
for all
z
E
Cx
and for all sET.
So every coset
Cx
is uniquely determined by the sequence of elements of G:
(5)
S(x)
= (
¢s(x)
I
sET) ,
the so-called syndrome of x (or of t.he coset containing x).
Let x
E
A
with syndrome (
f3s
I
s
E
T ) . To find the number of codewords
of weight
A
in
Cx
consists in solving the following problem: find the number of
codewords z
=
LgEG
z9
X9
of weight A satisfying
L
Zg98
=
/38
,
V
S
E
T .
gEG
When p
=
2 we obtain a system of diagonal equations over the finite field of
order
2m;
the problem consists in finding the number of solutions
satisfying
.X
(6)
L
Xi
=
/38
,
V
S
E
T .
i=O
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