Contemporary Mathematics
Volume 168, 1994
Tools for coset weight enumerators of some codes
PASCALE CHARPIN
ABSTRACT. Every extended primitive code C can be viewed in a group
algebra
K[G,
+],
where
K
and
G
are finite fields of same characteristic. Our
purpose is to show that the use of the multiplication of these algebra can
provide, in some situations, some tools which apply to the determination
of weight distributions of cosets of codes. Actually we will explain two
formulae which provide relations between the set of elements orthogonal
to a codeword x and the values of products xy, y E C.l. We give some
applications.
Keywords: cosets, cyclic codes, group algebra, equations on finite fields
1.
Introduction
The p-ary Reed-Muller codes (RM-codes) can be seen as polynomials codes
or as extended cyclic codes
[14].
Moreover BERMAN
[4]
proved that they are
the powers of the radical of the group algebra
A
=
K[ {
G,
+}] ,
K
=
G F(p)
and
G
=
GF(pm)
(m 1,
p
a prime). The Reed-Muller codes have remarkable
properties and all results on them involve results on some codes of length
pm
over K, the so-called extended primitive codes. For instance KASAMI deduced
weight distributions of two and triple error-correcting binary BCH codes from
the weight distributions of binary Reed-Muller codes of order 1 and 2
[15].
Set N
=
pm.
In this paper we treat only linear codes of length N over
K.
Such a code
C
is viewed as a K-subspace of
A
and we are mainly interested
by the weight distributions of cosets of C. Let D be the code generated by C
and a coset x
+
C of C. We will assume that the weight distribution of C.L is
known. As c
c
D ' D.L
c
c.L so that the weight distribution of
X
+
c can
be deduced from the weight distribution of C.L and of D.L. Hence the problem
consists of the determination, for any nonzero weight ,\ of C.L, of the number
of elements
y
of C.L of weight ,\ which are orthogonal to x. We will study the
1991Mathematics Subject Classification, Primary 94B, 12E20
This paper is in final form and will not be submitted for publication elsewhere.
©
1994 American Mathematical Society
0271-4132/94 Sl.OO
+
$.25 per page
http://dx.doi.org/10.1090/conm/168/01684
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