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

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