TOOLS FOR COSET WEIGHT ENUMERATORS 3
of
A:
{0}
=
pm(p-1)+1
C
pm(p-1)
C ... C
p2
C
p'
where
pm(p-
1)
= {
a1
I
a E
K }.
The position of an element or a subset of
A
in this sequence will appear as a useful parameter we define now:
DEFINITION
1. Let x
E
A; let U be a subset of
A.
Let
j
E
[0,
m(p-
1)].
We will say that the depth of x equals
j
if and only if x is in PJ and not
in
PH
1
.
We will say that the depth of U equals
j
if and only if U is included in
PJ and not included in pH1
.
By convention, P
0 =
A.
Let R be the quotient algebra K[ZJI(zn - 1) , where n
=
pm - 1 . A
cyclic code C* of length n over K is a principal ideal of R. It is generated by
a polynomial of K[Z] whose roots are called the zero's of C*. We consider the
extension C of the code C* in the algebra
A.
Let
a
be a primitive root of unity
in
G.
The extension is the usual one : each codeword c* E 'R, c*
=
E~
01
c;
zi,
is as follows extended to c E
A .
n-1 n-1
c
=
coX0
+
L
Ca;XQ; ,
Co=
-(L:
C:),
Cai
=
ci .
i=O i=O
Now consider the set
I
of those
k
such that
ak
is a zero of the cyclic code C*.
Then the codeword c is an element of C if and only if it satisfies:
n-1
L:c9 =0
and
LCa;(ak)i=O,
forallkEI.
gEG i=O
Therefore we can identify precisely an extended cyclic code in
A.
DEFINITION
2. Let S
=
[0,
n].
An extended cyclic code C in
A
is uniquely
defined by a subset T of S such that 0
E
T and T is a union of cyclotomic cosets
of p modulo n. Let us define for all s
E
S:
c/s :
X
E
A
f------+
L
x
9
g8
E
G .
gEG
In particular, ¢o(x)
=
LgEG
x
9

Then we have that
C
= {
x
E
A
I
¢s(x)
=
0 , for all sET }
We say that T is the defining set of the code C.
The p-ary Reed-Muller codes are extended cyclic codes in
A
[14].
Their
defining-sets are related with the p-weights of the elements of the interval S
=
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