4

PASCALE CHARPIN

[0,

n]:

for any s

E

S, let I.::,~ 1 sipi , Si

E

[O,p-

1] , be the p-ary expansion of

s;

then the p-weight of

s

is

m-1

wp(s) =Lsi.

i=O

DEFINITION

3. The p-ary Reed-Muller code of order r, denoted by Rp(r, m),

is the extended cyclic code in A, whose defining-set is

lp(j, m)

= {

s

E

S

I

wp(s)

j } , j

=

m(p- 1)- r .

BERMAN

proved in

[4]

that the p-ary Reed-Muller codes are the powers of the

radical of

A.

More precisely:

(3)

pi

=

Rp(m(p- 1) -

j,

m) , for all

j

E

[1, m(p- 1)]

Therefore, we can deduce from a wellknown property of the generalized Reed-

Muller codes that (Pi)j_

=

pm(p- 1)-i+ 1

.

In the following we will identify the

p-ary RM-codes with the codes pi. Note that

(4)

dim pi =card { s

E

S

I

wp(s)

?_

j }

Cosets of extended cyclic codes and equations over finite fields: Let

C

be an extended cyclic code with defining set

T.

Let x

E

A

and consider the

coset

Cx

=

x

+

C .

We have

¢s(z)

=

¢s(x) ,

for all

z

E

Cx

and for all sET.

So every coset

Cx

is uniquely determined by the sequence of elements of G:

(5)

S(x)

= (

¢s(x)

I

sET) ,

the so-called syndrome of x (or of t.he coset containing x).

Let x

E

A

with syndrome (

f3s

I

s

E

T ) . To find the number of codewords

of weight

A

in

Cx

consists in solving the following problem: find the number of

codewords z

=

LgEG

z9

X9

of weight A satisfying

L

Zg98

=

/38

,

V

S

E

T .

gEG

When p

=

2 we obtain a system of diagonal equations over the finite field of

order

2m;

the problem consists in finding the number of solutions

satisfying

.X

(6)

L

Xi

=

/38

,

V

S

E

T .

i=O

PASCALE CHARPIN

[0,

n]:

for any s

E

S, let I.::,~ 1 sipi , Si

E

[O,p-

1] , be the p-ary expansion of

s;

then the p-weight of

s

is

m-1

wp(s) =Lsi.

i=O

DEFINITION

3. The p-ary Reed-Muller code of order r, denoted by Rp(r, m),

is the extended cyclic code in A, whose defining-set is

lp(j, m)

= {

s

E

S

I

wp(s)

j } , j

=

m(p- 1)- r .

BERMAN

proved in

[4]

that the p-ary Reed-Muller codes are the powers of the

radical of

A.

More precisely:

(3)

pi

=

Rp(m(p- 1) -

j,

m) , for all

j

E

[1, m(p- 1)]

Therefore, we can deduce from a wellknown property of the generalized Reed-

Muller codes that (Pi)j_

=

pm(p- 1)-i+ 1

.

In the following we will identify the

p-ary RM-codes with the codes pi. Note that

(4)

dim pi =card { s

E

S

I

wp(s)

?_

j }

Cosets of extended cyclic codes and equations over finite fields: Let

C

be an extended cyclic code with defining set

T.

Let x

E

A

and consider the

coset

Cx

=

x

+

C .

We have

¢s(z)

=

¢s(x) ,

for all

z

E

Cx

and for all sET.

So every coset

Cx

is uniquely determined by the sequence of elements of G:

(5)

S(x)

= (

¢s(x)

I

sET) ,

the so-called syndrome of x (or of t.he coset containing x).

Let x

E

A

with syndrome (

f3s

I

s

E

T ) . To find the number of codewords

of weight

A

in

Cx

consists in solving the following problem: find the number of

codewords z

=

LgEG

z9

X9

of weight A satisfying

L

Zg98

=

/38

,

V

S

E

T .

gEG

When p

=

2 we obtain a system of diagonal equations over the finite field of

order

2m;

the problem consists in finding the number of solutions

satisfying

.X

(6)

L

Xi

=

/38

,

V

S

E

T .

i=O