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

=

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

=