MacWilliams Identities for Group Codes

In particular, we might assume that

m_i=p^e₁ⁱ¹p^e₂ⁱ². . . p^e_k^ik (3.22)

with p₁ > p₂ >· · · > p_k primes, and e_1j ≥ e_2j ≥ · · · ≥ e_rj ≥0 for all j = 1,2, . . . , k. We first define an inner product on Gⁿ:

Definition 3.5. Define a symmetric function< ., . >:Gⁿ×Gⁿ→Z_m1 such that

<(g₁, g₂, . . . , g_n),(h₁, h₂, . . . , h_n)>= Xn

i=1

g_i∗h_i (3.23)

where

g_i∗h_i =g_i(1)h_i(1) + m₁

m₂g_i(2)h_i(2) +· · ·+m₁

m_rg_i(r)h_i(r). (3.24) Here g_i(j)∈Z_mj is the jth coordinate of g_i when it is written as an r-tuple in G.

Note that the inner product thus defined is symmetric and bilinear. It is a natural definition for an inner product, because if ξ is a primitive m^st₁ root of unity over complex numbers then, ξ^m1/mi is a primitivem^th_i root of unity and hence a character of Z_mi. Definition 3.6. Suppose C is an abelian group code over G defined above by (3.21) of length n. We define thedual C^∗ of C with respect to the inner product defined above in (3.23) and (3.24) as

C^∗ =©

x∈Gⁿ¯

¯< x, c >= 0, ∀c∈Cª

. (3.25)

We prove the following lemma for the inner product above that will be useful:

Lemma 3.7. Suppose (h₁, h₂, . . . , h_n)∈Gⁿ is fixed. If

<(g₁, g₂, . . . , g_n),(h₁, h₂, . . . , h_n)>= 0

for all (g₁, g₂, . . . , g_n)∈Gⁿ, then (h₁, h₂, . . . , h_n) = 0, the zero vector in Gⁿ.

Proof. By taking all but one ofg_i’s to be the zero element in G, we can reduce this to the

casen= 1.So, supposeh∈Gis fixed and that< g, h >= 0 for allg∈G. This means that

g∗h=g(1)h(1) +m₁

m₂g(2)h(2) +· · ·+m₁

m_rg(r)h(r) = 0

for all g ∈G where g = (g(1), . . . , g(r)) withg(j) ∈Z_mj.Then take g = (1,0, . . . ,0). We see that we geth(1) = 0 inZ_m1. Takingg= (0,1,0, . . . ,0) we see that ^m_m¹

2h(2) = 0 inZ_m1, which means that h(2) = 0 in Z_m2. Similarly it can be shown that h(j) = 0 in Z_mj for j= 1,2, . . . , r and so we geth= 0 inG.

Remark 3.8. Lemma 3.7 implies that {0}^∗ =Gⁿ and (Gⁿ)^∗={0}.

To prove a MacWilliams-like identity for the complete weight enumerators of abelian group codes, we first prove the following lemma:

Lemma 3.9. Suppose H is a nontrivial subgroup ofZ_m1 of order s >1 and supposeξ is a primitive m^th₁ root of unity. Then we have

X

h∈H

ξ^h= 0.

Proof. Suppose 06=h∈H is an element in H of order s. Then

H ={0, h,2h, . . . ,(s−1)h}

withsh=m₁kfor some positive m₁.But then we have X

h∈H

ξ^h = Xs−1

`=0

(ξ^h)^` = ξ^sh−1 ξ^h−1 = 0

sinceξ^sh=ξ^m1k= 1.

We are now ready to prove the following theorem that gives a MacWilliams-like identity for the complete weight enumerators of abelian group codes:

Theorem 3.10. Suppose that G ={g₁, . . . , g_q} is an abelian group of order q and of the form given in (3.21). Suppose thatξ is a primitive m^th₁ root of unity over complex numbers,

and let C be a group code of length n over G, and let C^∗ be the dual of C with respect to the inner product defined in (3.23) and (3.24). Then we have

cwe_C^∗(W₁, W₂, . . . , W_q) = 1

|C|cwe_C¡X^q

i=1

ξ^g1∗giW_i, Xq

i=1

ξ^g2∗giW_i, . . . , Xq

i=1

ξ^gq∗giW_i¢

. (3.26)

Proof. The proof uses the same ideas and the same techniques used to prove the original MacWilliams theorem from [6] and [20]. We will first introduce a function overGⁿ as

F(u) := X

v∈Gⁿ

ξ<u,v>

Yq

i=1

W_i^ngi(v). (3.27)

Summing F(u)’s over all the codewords of the code C, we obtain X

u∈C

F(u) =X

u∈C

X

v∈Gⁿ

ξ<u,v>

Yq

i=1

W_i^ngi(v)= X

v∈Gⁿ

Yq

i=1

W_i^ngi(v)X

u∈C

ξ<u,v>. (3.28)

Now, suppose for fixedv∈Gⁿ, we consider the function

f_v :C→Z_m1

that takesu∈C to< u, v >modulom₁.Note thatf_v is a group homomorphism. Now, by definition of the dual, we have

ker(f_v) =C ⇔< u, v >≡0 (mod m₁) ∀u∈C ⇔v∈C^∗.

This means the inner sum of (3.28) becomes|C|for allv∈C^∗.

Now, suppose thatv is not inC^∗.This means that ker(f_v)6=C and so it is a non-trivial subgroup of C, which means that Im(f_v) is a non-trivial subgroup of Z_m1 and hence by Lemma 3.9, the inner sum becomes 0 for any such v∈Gⁿ. This means that

X

u∈C

F(u) =|C| X

v∈C^∗

Yq

i=1

W_i^ngi(v) =|C|cwe_C^∗(W₁, W₂, . . . , W_q), (3.29)

which is equivalent to saying that

cwe_C^∗(W₁, W₂, . . . , W_q) = 1

|C|

X

u∈C

F(u). (3.30)

Now we need to find whatF(u) is. Letδ(x, y) denote the Kronecker Delta function, which takes the value 1 ifx=y and 0 for all other values. So

F(u) = X

v∈Gⁿ

ξ<u,v>

Yq

i=1

W_i^ngi(v)

= X

(v1,v2,...,vn)∈Gⁿ

µY_n

j=1

¡ξ^uj∗vj Yq

i=1

W_i^δ(vj,gi)¢¶

= Yn

j=1

µXq

i=1

ξ^uj∗giW_i

¶

= µX_q

i=1

ξ^g1∗giW_i

¶_ng

1(u)

· µX_q

i=1

ξ^g2∗giW_i

¶_ng

2(u)

· · · · µX_q

i=1

ξ^gq∗giW_i

¶_ngq(u) .

Summing this last product over all the codewords ofC, we get the desired result.

As we said earlier, knowing the complete weight enumerator makes it easier to calculate all the other weight enumerators. Now, as an application we will obtain the MacWilliams identity for the Hamming weight enumerators of abelian group codes by using Theorem 3.10, which will be the exact same result that Delsarte obtained in [5].

MacWilliams identity for Hamming weight enumerators For this section, we will assumeG=©

g₁, g₂, . . . , g_qª

to be the abelian group over which the code is defined and g₁ = 0 the zero element in the group. We first define the Hamming weight enumerator for the group code in a similar way that it was defined forZ₄-codes:

H_C(W, X) =X

c∈C

W^n−wH(c)X^wH(c) =X

c∈C

W^n0(c)X^{ng2(c)+···+ngq(c)} (3.31)

where w_H is the Hamming weight. In fact, we note that it can be written in terms of the complete weight enumerator as

H_C(W, X) = cwe_C(W, X, X, . . . , X). (3.32)

Then using Theorem 3.10, we can relate the Hamming weight enumerator ofC and C^∗:

H_C^∗(W, X) = cwe_C^∗(W, X, X, . . . , X)

= 1

|C|cwe_C¡

W + (q−1)X, W +X Xq

i=2

ξ^g2∗gi, . . . , W +X Xq

i=2

ξ^gq∗gi¢ . (3.33)

At this point we want to introduce the following lemma:

Lemma 3.11. For all j >1, we have Xq

i=2

ξ^gi∗gj =−1.

Proof. For any such fixedj, consider the mapf_j :G→Z_m1 such thatf_j(g_i) =g_i∗g_j. Then f_j is a group homomorphism. By Lemma 3.7, we know thatg_i∗g_j = 0 for allj= 1,2, . . . , q if and only if g_i =g₁ = 0. So, if j > 1, then we have ker(f_j) 6=G and hence Im(f_j) is a non-trivial subgroup ofZ_m1. But then, by Lemma 3.9 we get

Xq

i=1

ξ^gj∗gi = 0

for all such j >1.Since g_j∗g₁=g_j ∗0 = 0 for all j, we get Xq

i=2

ξ^gi∗gj =−1, j = 2,3, . . . , q.

But using Lemma 3.11 in (3.33), we see that we have easily proved the following corollary of Theorem 3.10 that gives the MacWilliams identity for the Hamming weight enumerators of abelian group codes:

Corollary 3.12. Suppose that C is a group code overG of length nwith |G|=q. Let

H_C(W, X) =X

c∈C

W^n−wH(c)X^wH(c)

be the Hamming weight enumerator of C and let C^∗ be the dual of C with respect to the inner product defined in (3.23) and (3.24). Then we have the Macwilliams identity for the Hamming Weight enumerators ofC and C^∗ as follows:

H_C^∗(W, X) = 1

|C|H_C¡

W + (q−1)X, W−X¢ .