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

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