TOOLS FOR COSET WEIGHT ENUMERATORS 11

2) Suppose that a code C is such that its dual is a union of L cosets

y

+

pm(p-

1)-r+1

,

y

E

pm(p-

1)-r

,

where all cosets have the same weight enumer-

ator. Then there is only one weight distrihution for the cosets x

+

C, x

E

pr

and this result holds for non extended cyclic codes

C.

It becomes from the fact

that L/2 cosets composing Cl. are orthogonal to x, for any x. We will treat later

a special application of this fact (see Corollary 3).

3.4. Application of Formula (II). In this section we will consider ex-

tended cyclic codes C of depth r, for some

r,

such that:

pr+1

c

c

c

pr (

i.e.

pm(p-1)-r+1

c

cl.

c

pm(p-1)-r) .

Let Tl. be the defining set of the dual code Cl.. In accordance with Definition

3 and with the definition of C, Tl. has the following form:

{13)

Tl.

=

Ip(m(p-1) -r+ 1,m) \

{s1, ... ,s1},

where the

Si

are some elements of S of p-weight m(p-

1) - r.

Note that the

definingsetofCis T=Ip(r,m)U{n-sl···

,n-sz}.

Set

U={s1,··· ,sz}

and let E be the cyclic code (in the algebra R) whose nonzeros are the ol,

t

E U. The code E can be viewed in the algebra A , by adding an all-zero

column to the generator matrix. Clearly depth( E) equals depth( Cl.) equals

m(p- 1) -

r.

Then, since the dimension of E equals the number of elements of U,

which is the dimension of the quotient space cl.

;pm(p-

1)-r+1

,

we have clearly

{14)

cJ.

=

u

(y

+

pm(p-1)-r+l) .

yEE

The definitions of Cl., U and E hold in all this section. Recall that Dx is the

linear code generated by C and the coset of C containing

x;

we want to determine

the weight distribution of the dual of Dx·

PROPOSITION

5. Let a coset of C: Cx

=

x

+

C , x

E

pr \ C .

Let yin

E.

Then, for every

j

E [O,n -1]:

l

¢n(x aj(y))

=

0 if and only if L¢n-s,(x)¢s,(y)ast1

=

0.

t=1

That means that the number of shifts of y orthogonal to x is related with the

weight of the codeword y' of E defined by

{15)

Let n(y) be the number of elements of the set { aJ(Y)

I

j E

[0,

n-

1] }.

Then we

have

n(y) '

L(y) =card { a1(y)

I

x, aj(y)

=

0}

=

-(n- w(y)) ;

n

the code

v;

contains L(y) cosets a1(y)

+

pm(p-l)-r+l.

2) Suppose that a code C is such that its dual is a union of L cosets

y

+

pm(p-

1)-r+1

,

y

E

pm(p-

1)-r

,

where all cosets have the same weight enumer-

ator. Then there is only one weight distrihution for the cosets x

+

C, x

E

pr

and this result holds for non extended cyclic codes

C.

It becomes from the fact

that L/2 cosets composing Cl. are orthogonal to x, for any x. We will treat later

a special application of this fact (see Corollary 3).

3.4. Application of Formula (II). In this section we will consider ex-

tended cyclic codes C of depth r, for some

r,

such that:

pr+1

c

c

c

pr (

i.e.

pm(p-1)-r+1

c

cl.

c

pm(p-1)-r) .

Let Tl. be the defining set of the dual code Cl.. In accordance with Definition

3 and with the definition of C, Tl. has the following form:

{13)

Tl.

=

Ip(m(p-1) -r+ 1,m) \

{s1, ... ,s1},

where the

Si

are some elements of S of p-weight m(p-

1) - r.

Note that the

definingsetofCis T=Ip(r,m)U{n-sl···

,n-sz}.

Set

U={s1,··· ,sz}

and let E be the cyclic code (in the algebra R) whose nonzeros are the ol,

t

E U. The code E can be viewed in the algebra A , by adding an all-zero

column to the generator matrix. Clearly depth( E) equals depth( Cl.) equals

m(p- 1) -

r.

Then, since the dimension of E equals the number of elements of U,

which is the dimension of the quotient space cl.

;pm(p-

1)-r+1

,

we have clearly

{14)

cJ.

=

u

(y

+

pm(p-1)-r+l) .

yEE

The definitions of Cl., U and E hold in all this section. Recall that Dx is the

linear code generated by C and the coset of C containing

x;

we want to determine

the weight distribution of the dual of Dx·

PROPOSITION

5. Let a coset of C: Cx

=

x

+

C , x

E

pr \ C .

Let yin

E.

Then, for every

j

E [O,n -1]:

l

¢n(x aj(y))

=

0 if and only if L¢n-s,(x)¢s,(y)ast1

=

0.

t=1

That means that the number of shifts of y orthogonal to x is related with the

weight of the codeword y' of E defined by

{15)

Let n(y) be the number of elements of the set { aJ(Y)

I

j E

[0,

n-

1] }.

Then we

have

n(y) '

L(y) =card { a1(y)

I

x, aj(y)

=

0}

=

-(n- w(y)) ;

n

the code

v;

contains L(y) cosets a1(y)

+

pm(p-l)-r+l.