Complete Joint Weight Enumerators and Self-Dual Codes YoungJu Choie ∗ Department of Mathematics Pohang University of Science and Technology Pohang, 790-784, Korea Email: [email protected] Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA Haesuk Kim † Department of Mathematics Pohang University of Science and Technology Pohang, 790-784 Korea June 22, 2011

Abstract We define the complete joint weight enumerator in genus g for codes over Z2k and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series.

∗

This research was partially supported by Com2 MaC-KOSEF and KOSEF2000-015-DP0009. † This research was supported by Com2 MaC-KOSEF

1

Keywords: Self-Dual Codes, Unimodular Lattices, Jacobi and Siegel Forms.

Contact author: Steven Dougherty Address: Department of Mathematics University of Scranton Scranton, PA 18510 Telephone: 570-941-6104 Fax: 570-941-5981 E-mail: [email protected]

2

1

Introduction

A code C of length n over Z2k is a subset of Zn2k , and if it is an additive subgroup then we say that it is a linear code. We shall use the following weights on this ambient space. Let v = (v1 , . . . , vn ) be a vector in Zn2k , then the Hamming weight wtH (v) is the number P of non-zero components and the Euclidean weight wtE (v) = ni=1 min{vi2 , (2k − vi )2 }. The minimum Hamming and Euclidean weights are denoted by dH and dE respectively, and are the smallest Hamming and Euclidean weights among all non-zero codewords of C. We use the standard inner product of x and y in Zn2k , specifically [x, y] = x1 y1 +· · ·+xn yn (mod 2k) where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). The dual code C ⊥ of C is defined as C ⊥ = {x ∈ Zn2k | [x, y] = 0 for all y ∈ C}. We say that a code C is self-orthogonal if C ⊆ C ⊥ and self-dual if C = C ⊥ . In [1] Type II codes over Z2k are defined as self-dual codes with Euclidean weights divisible by 4k, and those self-dual codes which are not Type II are said to be Type I. This definition is natural because Type II codes produce Type II lattices and Type I codes produce Type I lattices under the usual construction. See equation (1) below. We shall take the standard definition of a shadow of a self-dual code. Namely, let C be a Type I code over Z2k . Define C0 as the subcode of vectors whose Euclidean weight is congruent to 0 (mod 4k). Then C0 is index 2 in C and C0⊥ = C ∪ S where S is the shadow and C = C0 ∪ C2 and S = C1 ∪ C3 . P Let Rn be n-dimensional Euclidean space with inner product x · y = xi yi . An n n dimensional lattice L in R is a free Z-module spanned by n linearly independent vectors. The dual lattice is defined by: L∗ = {x ∈ Rn | x · v ∈ Z for all v ∈ L}. A lattice is integral if L ⊆ L∗ and unimodular if L = L∗ . A unimodular lattice is Type II if the norms of all elements are even and the lattice is Type I otherwise, where the norm of a vector v is N (v) = v · v. Define the reduction modulo 2k, by ρ : Zn → Zn2k , by ρ(x1 , . . . , xn ) = (x1

(mod 2k), . . . , xn

(mod 2k)).

Given a code C over Z2k we construct a lattice by (1)

1 Λ(C) = √ {x ∈ Zn | ρ(x) ∈ C} 2k

It is shown in [1] that if C is a Type I code then Λ(C) is a Type I unimodular lattice, and that if C is a Type II code then Λ(C) is a Type II unimodular lattice. As for codes we make an analogous definition of the shadow of a lattice. Let L be a Type I lattice and let L0 denote the subset of vectors with even norms. Then L0 is index 2 in L 3

and L∗0 = L ∪ Σ where Σ is the shadow and L = L0 ∪ L2 and Σ = L1 ∪ L3 . It is shown in [8] that (2) Λ(Ci ) = Λ(C)i for i = 0, 2 and for i = 1, 3 up to labeling. In particular Λ(S) = Σ where Σ is the shadow of the unimodular lattice Λ(C).

2

Weight Enumerators

Throughout the remainder of the paper we shall often denote the ring Z2k by R. The following definition is a generalization of the complete weight enumerator in genus g introduced in [1] and of the standard joint weight enumerator. Definition 1 (Complete Joint Weight Enumerator in Genus g) Let C1 , C2 , .., Cg be codes in Rn . The complete joint weight enumerator for codes C1 , . . . , Cg of length n over R is defined as (3)

JC1 ,C2 ,...,Cg (Xa with a ∈ Rg ) =

X

Y

Xana (c1 ,c2 ,...,cg )

(c1 ,c2 ,...,cg )∈C1 ×C2 ×...×Cg a∈Rg

where na (c1 , c2 , . . . , cg ) = |{j|((c1 )j , (c2 )j , . . . , (cg )j ) = a}|, and ci = ((ci )1 , . . . , (ci )n ). Note that we do not assume that the codes are linear, in fact we are interested in determining complete joint weight enumerators where some of the codes are shadows, which are non-linear codes. We shall often denote the complete joint weight enumerator by JC1 ,...,Cg (Xa ). Notice that SC,g (Xa ) = JC,C,...,C (Xa ), where SC,g (Xa ) is the complete weight enumerator for a code C given in [1]. Notice however that the MacWilliams relations given in [1] apply only in this specific case. MacWilliams relations for the complete weight enumerator in genus g must be able to determine JD1 ,...,Dg (Xa ) from JC1 ,...,Cg (Xa ), where Di is Ci for some i and Di is Ci⊥ for other i. These MacWilliams relation will be given in Section 2.1. The complete joint weight enumerator in genus g is also related in a natural way to the following weight enumerators. Let Rg be the set of equivalence classes of Rg where the elements a, b ∈ Rg are equivalent if a = b or a = −b. Then we can make the following definition. Definition 2 (Symmetrized Joint Weight Enumerator in Genus g) Let C1 , .., Cg be codes in Rn . The symmetrized joint weight enumerator for codes C1 , . . . , Cg of length n over R is defined as (4)

X

MC1 ,C2 ,...,Cg (Xa with a ∈ Rg ) =

Y

(c1 ,c2 ,...,cg )∈C1 ×C2 ×...×Cg a∈Rg

4

n (c1 ,c2 ,...,cg )

Xa a

.

Definition 3 (Joint Weight Enumerator in Genus g) Let C1 , C2 , . . . , Cg be codes in Rn . The joint weight enumerator for codes C1 , . . . , Cg of length n over R is defined as HC1 ,C2 ,...,Cg (Xa with a ∈ Zg2 ) =

(5)

X

Y

Xaha (c1 ,c2 ,...,cg )

(c1 ,c2 ,...,cg )∈C1 ×C2 ×...×Cg a∈(Z2 )g

with ha (c1 , c2 , . . . , cg ) = |{j|((c1 )j , (c2 )j , . . . , (cg )j ) ≡ a}|, where an element a ≡ 1 if and only if a 6= 0. The joint weight enumerator in genus g was introduced in [7] and is the natural generalization of the joint weight enumerator. Q Let C1 , C2 , . . . , Cg be codes over Z2k of length n. Then if P (Xa ) = a∈A (Xa ), where A a subset of Zg2k is a monomial in JC1 ,C2 ,...,Cg (Xa ) and P (Xa ) represents vectors c1 , . . . , cg then by counting the weights of c1 , . . . , cg in different orders we have X

wtH (ci ) =

X

wtH (a) and

X

wtE (ci ) =

a∈A

X

wtE (a).

a∈A

Of course, a similar theorem is true for the symmetrized joint weight enumerator and the Euclidean weight and for the joint weight enumerator and the Hamming weight. Q It follows then that for all monomials P (Xa ) = a∈A (Xa ) in JC1 ,C2 ,...,Cg (Xa ) we have X

wtH (a) ≥

(dH )i

i=1

a∈A

X

g X

wtE (a) ≥

g X

(dE )i .

i=1

a∈A

Again similar results can be made for the other weight enumerators.

2.1

MacWilliams Relations

We shall describe the following notation to describe the action of a matrix on a polynomial ring. Let A = (aij ) be an n by n matrix and f (x1 , . . . , xn ) a polynomial in C[x1 , x2 , . . . , xn ] then X X (6) A · f (x1 , . . . , xn ) = f ( a1j xj , . . . , anj xj ). 1≤j≤n

1≤j≤n 2πi

Let η2k be a primitive 2k-th root of unity, that is η2k = e 2k . The matrix T is given by: (7)

ab Ta,b = η2k

where a and b are in R.

5

Theorem 2.1 (MacWilliams Relations) Let C1 , C2 , . . . , Cg be codes in Rn and let C˜ denote either C or C ⊥ . Then (8)

1

JC˜1 ,C˜2 ,...,C˜g (Xa ) = Qg

i=1

where

(

δC˜ =

δC˜

· (⊗gi=1 T

if if

C˜ = C, C˜ = C ⊥ .

|Ci | 0 1

δC˜

i

)JC1 ,...,Cg (Xa )

i

Proof. The proof of this theorem follows exactly as the proof given for the joint weight enumerator in [7], except that the function na is changed and the matrix is changed accordingly. 2 Note that the matrix ⊗gi=1 T polynomial in |R|g variables.

δC˜

i

is an |R|g by |R|g matrix and that JC˜1 ,C˜2 ,...,C˜g (Xa ) is a

Corollary 2.2 Let C1 , C2 , . . . , Cg be codes in Rn and let C˜ denote either C or C ⊥ . Then (9)

1

MC˜1 ,C˜2 ,...,C˜g (Xa ) = Qg

δC˜

i=1

|Ci |

X

0 Ta,d

· (Tg )MC1 ,...,Cg (Xa )

i

where (Tg )a,b =

a, b ∈ Rg

d=±b

with T 0 = ⊗gi=1 T and (10)

HC˜1 ,C˜2 ,...,C˜g (Xa ) = Qg

i=1

1 δC˜

|Ci |

δC˜

i

· (⊗gi=1 H

where H=

δC˜

i

)HC1 ,...,Cg (Xa )

i

1 2k − 1 1 −1

!

Proof. The result follows from the previous theorem.

2

Note that the MacWilliams relation for H were given in [7]. These are not the only possible weight enumerators that can be obtained from the complete joint weight enumerator. For example, two elements can be related in a variety of ways, see [13] for examples.

6

3

Joint Weight Enumerators of Shadows and Codes

In this section we show how the complete joint weight enumerator relating shadows and Type I codes can be determined. Let C1 , C2 , . . . , Cg be Type I codes in Zn2k . Let (Ci )0 denote the subcode of Ci consisting of those vectors that have Euclidean weight congruent to 0 (mod 4k). For a code Ci define its shadow by Si = (Ci )⊥ 0 − Ci , with Si = (Ci )1 ∪ (Ci )3 . Let I be a subset of {1, 2, . . . , g}. Define φI as following: P

φI (a) =

(11)

( η4k

i∈I

a2i )

for a ∈ Rg . Theorem 3.1 Let C1 , . . . , Cg be Type I codes over R then (12)

JC10 ,C20 ,...,Cg0 (Xa ) =

where

(

Ci0

=

1 X JC ,C ,...,Cg (φJ (a)Xa ) 2|I| J⊆I 1 2 Ci (Ci )0

if i ∈ / I, if i ∈ I.

Proof. The proof proceeds by induction on the cardinality of I. Let I = {i1 , . . . , ik }. (13)

1 JC1 ,...,(Ci1 )0 ,...,Cg (Xa ) = (JC1 ,C2 ,...,Cg (Xa ) + JC1 ,C2 ,...,Cg (φ{i1 } (a)Xa )) 2

since any monomial P (Xa ) representing vectors v1 , v2 , . . . , vg such that vi1 ∈ (Ci1 )0 has P (φ{i1 } (a)Xa ) = P (Xa ) and P (φ{i1 } (a)Xa ) = −P (Xa ) otherwise. Next assume that equation (12) holds for {i1 , . . . , i`−1 }. (14)

1 JC10 ,...,(Ci` )0 ,...,Cg0 (Xa ) = (JC10 ,C20 ,...,Cg0 (Xa ) + JC10 ,C20 ,...,Cg0 (φ{i` } (a)Xa )) 2

where

(

Ci0

=

Ci (Ci )0

if i ∈ / {i1 , . . . , i`−1 }, if i ∈ {i1 , . . . , i`−1 }.

Then notice that φ{i} (a)φA (a) = φA∪{i} (a) for i ∈ / A, and we have the result. Let Si denote the shadow of the Type I code Ci , i.e. Si = (Ci )1 ∪ (Ci )3 . Theorem 3.2 Let C1 , . . . , Cg be Type I codes over R then (15)

1  ⊗gi=1 T Cbi JC1 ,C2 ,...,Cg (φI (a)Xa )  C b i i=1 |Ci |

JCc1 ,Cc2 ,...,Ccg (Xa ) = Qg

7

2

where

(

Cbi = and

( c C

i

=

0 1

if i ∈ / I, if i ∈ I,

Ci Si

if i ∈ / I, if i ∈ I.

Proof. Again we proceed by induction on the cardinality of I. Let I = {i1 , . . . , ik } and let  Tα = ⊗gi=1 T Cbi with Cbi = 1 if and only if i = α. JCc1 ,...,Si

1

cg ,...,C

(Xa )

= JC1 ,...,(Ci1 )⊥0 ,...,Cg (Xa ) − JC1 ,C2 ,...,Cg (Xa ) 1 Ti · JC1 ,C2 ,...,(Ci1 )0 ,...,Cg (Xa ) − JC1 ,C2 ,...,Cg (Xa ) = |(Ci1 )0 | 1 1 = Ti · (JC1 ,C2 ,...,Cg (Xa ) + JC1 ,C2 ,...,Cg (φ{i1 } (a)Xa )) − JC1 ,C2 ,...,Cg (Xa ) 2|(Ci1 )0 | 1 1 Ti · (JC1 ,C2 ,...,Cg (φ{i1 } (a)Xa )). = |Ci1 | 1 Next assume that equation (15) holds for {i1 , . . . , i`−1 }. Specifically, (16)

1  ⊗gi=1 T Cbi JC1 ,C2 ,...,Cg (φ{i1 ,...,i`−1 } (a)Xa )  C b i i=1 |Ci |

JCc1 ,Cc2 ,...,Ccg (Xa ) = Qg

c = S if and only if i ∈ {i , . . . , i }. Then where C i i 1 `−1

JCc1 ,...,Si

`

cg ,...,C

(Xa )

= JCc1 ,...,(Ci )⊥ ,...,Ccg (Xa ) − JCc1 ,Cc1 ,...,Ccg (Xa ) ` 0 1 = Ti · (JCc1 ,Cc2 ,...,Ccg (Xa ) + JCc1 ,Cc2 ,...,Ccg (φ{i` } (a)Xa )) − JCc1 ,Cc2 ,...,Ccg (Xa ) |Ci` | ` 2

and the result follows as above.

A less general version of this theorem was stated incorrectly in [1]. Specifically, the constant was missing. Other weight enumerators can be computed in similar ways, for example, JC100 ,C200 ,...,Cg00 where Ci00 is either Ci , (Ci )0 or Si . Moreover each of these weight enumerators must have non-negative integral coefficients for all monomials in order for the code to exist. Corollary 3.3 Let C1 , . . . , Cg be Type I codes over R then (17)

MC10 ,C20 ,...,Cg0 (Xa ) =

1 X MC1 ,C2 ,...,Cg (φJ (a)Xa ) 2|I| J⊆I 8

where Ci0 is defined as in Theorem 3.1, and  1 C bi  Tg JC1 ,C2 ,...,Cg (φI (a)Xa ) C b i i=1 |Ci |

(18)

MCc1 ,Cc2 ,...,Ccg (Xa ) = Qg

c is defined as in Theorem 3.2. where C i

Proof. Follows from the previous two theorems noting that i2 = (−i)2 in Z2k .

2

In [9] it was shown that if C and D are Type I codes of length n and m respectively, with either n ≡ m ≡ 0 (mod 2) or n ≡ m ≡ 1 (mod 2) with n + m ≡ 0 (mod 4), then E = ∪3i=0 (Ci , Di )

(19)

is a self-dual code where C0 and D0 are the subcodes of vectors with weight congruent to 0 (mod 4k) and the other cosets are defined as usual. The code is said to be formed by the shadow sum of C and D. Theorem 3.4 Let C1 , C2 , . . . , Cg be self-dual codes of length n and D1 , D2 , . . . , Dg be selfdual code of length m with n and m satisfying the above conditions so that Ei is the self-dual code formed from the shadow sum of Ci and Di . Then (20) JE1 ,E2 ,...,Eg (Xa ) =

3 X 3 X

...

i1 =0 i2 =0

3 X

J(C1 )i1 ,(C2 )i2 ,...,(Cg )ig (Xa )J(D1 )i1 ,(D2 )i2 ,...,(Dg )ig (Xa )

ig =0

Proof. The code Ej = ∪3i=1 ((Cj )i , (Dj )i ) so we have (21)

JE1 ,E2 ,...,Eg (Xa ) =

3 X 3 X i1 =0 i2 =0

3 X

...

J((C1 )i1 ,(D1 )i1 ),((C2 )i2 ,(D2 )i2 ),...,((Cg )ig ,(Cg )ig ) (Xa ).

ig =0

Then we have J

((C1 )i1 ,(D1 )i1 ),((C2 )i2 ,(D2 )i2 ),...,((Cg )ig ,(Cg )ig ) (Xa )

= J(C1 )i1 ,(C2 )i2 ,...,(Cg )ig (Xa )J(D1 )i1 ,(D2 )i2 ,...,(Dg )ig (Xa ) 2

and this gives the result.

3.1

Generalized Shadows

A generalized version of the shadow for codes over Z2k was introduced in [5]. Specifically, if C is a self-dual code over Z2k and s = (α, α, . . . , α) is a constant vector not in C, such that 2s ∈ C, then we define sC0 = {v ∈ C | [v, s] = 0 for all v ∈ C}. 9

Then sS = C + s so that sC0⊥ = C ∪ sS. In general s is chosen to be a constant vector so that all the respective weight enumerators can be computed. These shadows are useful for construction of larger self-dual codes as in Equation (19). Since (k, k, . . . , k) is in C for all self-dual codes over Z2k it is not difficult to produce such an s by simply dividing this vector by 2, until we have a vector that is not in C. Of course it is possible that all of these vectors are in C in which case such a shadow is simply equal to the code as the usual shadow is for a Type II code. Let C1 , C2 , . . . , Cg be self-dual codes in Zn2k and let s1 , s2 , . . . , sg be vectors such that si ∈ / Ci , 2si ∈ Ci and si = (αi , αi , . . . , αi ). We choose αi as described above so that αi divides 2k for all i. Let si (Ci )0 denote the subcode of Ci consisting of those vectors orthogonal to si . Define si Si as above. Let I be a subset of {1, 2, . . . , g}. Define ψI as following: ψI (a) =

(22)

Y i∈I

(η 2k )ai αi

for a = (a1 , . . . , ag ) ∈ Rg . Theorem 3.5 Let C1 , . . . , Cg be self-dual codes over R then (23)

JC10 ,C20 ,...,Cg0 (Xa ) =

where

(

Ci0 and (24)

=

1 X JC ,C ,...,Cg (ψJ (a)Xa ) 2|I| J⊆I 1 2 Ci si (Ci )0

if i ∈ / I, if i ∈ I.

1  ⊗gi=1 T Cbi JC1 ,C2 ,...,Cg (ψI (a)Xa )  C bi i=1 |Ci |

JCc1 ,Cc2 ,...,Ccg (Xa ) = Qg

where

(

Cbi = and

( c C

i

=

0 1

Ci si S i

if i ∈ / I, if i ∈ I, if i ∈ / I, if i ∈ I.

Proof. Similar to the proof of Theorem 3.1 and Theorem 3.2.

3.2

Invariant Rings for Shadow Codes

˜ 2k,g be a subgroup of GL((2k)g , C) defined as Let G ˜ 2k,g = h T2k,g (i), D2k,g (S) | 1 ≤ i ≤ g, S is an even integral symmetric matrix i G where

1 aSat T2k,g (i) = √ ⊗gj=1 T δij , D2k,g (S) = diag(η4k )a∈Zg2k . 2k 10

2

Theorem 3.6 Let C1 , C2 , .., Cg be self-dual codes in Zn2k . For codes C1 , .., Cg , the complete ˜ 2k,g . joint weight enumerator JC1 ,...,Cg is invariant under the action of the group G Proof. The fact that the weight enumerator is invariant under T2k,g (i), 1 ≤ i ≤ g, follows from Theorem 2.1. The invariance of D2k,g (S) can be shown in a similar manner to the proof of Theorem 5.3 in [1]. Notice that S runs over all even integral symmetric matrices. 2 g 1 |I| The matrices T2k,g (i), 1 ≤ i ≤ g, generate all matrices of the form T2k,g (I) = ( √2k ) ⊗i=1 T i ˜ 2k,g also contains all matrices of the form with i = 1 if and only if i ∈ I. Thus the group G T2k,g (I). These matrices represent the action of the MacWilliams relations on any subset of the indices. c be as in Theorem 3.2, specifTheorem 3.7 Let C1 , .., Cg be Type I codes over Z2k and C i c ically Ci is either the code or its shadow depending on i. Then the complete joint weight −1 ˜ c , .., C c, J enumerator for C 1 g c1 ,..,C cg is invariant under the action of the group A G2k,g A, C where 1  ⊗gi=1 T Cbi · diag(φI (a))a∈Zg2k . A= √ |I| ( 2k)

2

Proof. The result follows from Theorem 3.2 and Theorem 3.6.

Similarly, we define the invariant ring for the symmetrized joint weight enumerator in ˜ 2k,g be a subgroup of GL(2g−1 (k g + 1), C) defined by genus g. Let H ˜ 2k,g = h T 2k,g (i), D2k,g (S) | 1 ≤ i ≤ g, S is an even integral symmetric matrix i H where (T 2k,g (i))a,b =

X

(T2k,g )a,d

d=±b

t

aSa a, b ∈ Rg , D2k,g (S) = diag(η4k )a∈Zg . 2k

Theorem 3.8 Let C1 , C2 , .., Cg be self-dual codes in Zn2k . For codes C1 , .., Cg , the sym˜ 2k,g . metrized joint weight enumerator MC1 ,...,Cg is invariant under the action of the group H c Let C1 , .., Cg be Type I codes over Z2k and Ci be as in Theorem 3.2. Then the symmetrized c , .., C c, M joint weight enumerator for C 1 g c1 ,..,C cg is invariant under the action of the group C −1 ˜ A H 2k,g A, where Aa,b =

X

Aa,d

a, b ∈ Rg .

d=±b

Proof. Similar to the proof of Theorem 3.6.

11

2

3.3

Invariant Rings for Generalized Shadow Codes

In this subsection we consider the invariant rings for generalized shadow codes. From the fact that the complete joint weight enumerator for the generalized shadow codes can be obtained from the complete joint weight enumerator for codes, we easily obtain the following theorems. c be as in Theorem 3.5, Theorem 3.9 Let C1 , .., Cg be self-dual codes over Z2k and Ci 0 , C i c is either the code or its generalized shadow depending on i. For codes C c , .., C c, specifically C i 1 g the complete joint weight enumerator JCc1 ,...,Ccg is invariant under the action of the group ˜ 2k,g B, where B −1 G 1  B= √ ⊗gi=1 T Cbi · diag(ψI (a))a∈Zg2k . |I| ( 2k)

Proof. Similar to the proof of Theorem 3.7.

2

In a similar manner to Theorem 3.8, we have the following theorem. c , .., C c , the symTheorem 3.10 Let C1 , C2 , .., Cg be self-dual codes in Zn2k . For codes C 1 g −1 ˜ metrized joint weight enumerator MCc1 ,...,Ccg is invariant under the action of the group B H 2k,g B, where X B a,b = Ba,d a, b ∈ Rg . d=±b

Proof. Similar to the proof of Theorem 3.8.

4

2

Joint Weight Enumerators, Jacobi forms and Siegel modular forms

In this section we show the connection between the complete joint weight enumerators of codes over R and Jacobi forms. First, we recall the definition of Jacobi forms of higher genus given in [14]. Let Hg be the Siegel upper half space of genus g; Hg = {τ ∈ Mg×g (C) | τ t = τ, Im(τ ) > 0}. Let χ be a character on the sympletic group Sp2g (R). For a holomorphic function f : Hg × Cg → C and any nonnegative integer k and m, the slash operators with respect to χ are defined as (f |k,m M )(τ, z) = (

det(Cτ + D) −k −2πimz(Cτ +D)−1 Czt ) e f ((Aτ + B)(Cτ + D)−1 , z(Cτ + D)−1 ) χ(M )

12

and t

t

(f |m [λ, µ])(τ, z) = e2πim(λτ λ +2λz ) f (τ, z + λτ + µ), !

A B where M = ∈ Sp2g (R), λ, µ ∈ Rg . C D For a finite index subgroup Γ of Γg = Sp2g (Z) we define a Jacobi form as follows. Definition 4 (Jacobi form) Let k and m be nonnegative integers and χ be a character of Γ. A holomorphic function f : Hg × Cg → C satisfying (25)

(f |k,m M )(τ, z) = f (τ, z), for all M ∈ Γ,

(26)

(f |m [λ, µ])(τ, z) = f (τ, z), for all λ, µ ∈ Zg

and having a Fourier expansion of the following form : (27)

X

f (τ, z) =

t

c(N, R)e2πiσ(N τ ) e2πiRz ,

X

N =N t ≥0 R∈Zg

N 21 Rt is half integral positive semidefinite, is called a Jacobi with c(N, R) 6= 0 only if 1 R m 2 form of weight k and index m with respect to χ on Γ. Here, σ(x) is the trace of x. !

We denote the space of Jacobi forms of weight k and index m with respect to χ on Γ by Jk,m (Γ, χ). In the case when χ is trivial, we denote the space of Jacobi forms by Jk,m (Γ). Remark 1 For f ∈ Jk,m (Γ, χ), f (τ, 0) is a Siegel modular form of weight k of genus g with respect to χ on Γ. Let I be a subset of {1, 2, . . . , g} and C1 , .., Cg be self-dual codes over R. We denote the c as Λ c , where lattice induced from the code C i i c C

(

Ci if i ∈ / I, , Si if i ∈ I.

(

Λi if i ∈ / I, Σi if i ∈ I.

=

i

so that c Λ

i

=

Note that the lattice formed from the shadow of a Type I code is the shadow of the formed unimodular lattice(see equation (2)). Define a theta series ϑΛc1 ,..,Λcg on Hg × C(1,g) as followings; (28)

ϑΛc1 ,..,Λcg (τ, z) :=

X

t

eπiσ(xτ x )+2πi(

√

2k(1,..,1)xz t )

, τ ∈ Hg , z ∈ C(1,g) .

c1 ×··×Λ cg x∈Λ

Also note that if Ci is a Type II code over R then Si = Ci . The next theorem gives a connection between the theta series defined over the lattices induced from codes and their weight enumerators. 13

Theorem 4.1 Let C1 , .., Cg be self-dual codes over R. For the complete joint weight enuc , .., C c , we have merator JCc1 ,..,Ccg (Xa ) for C 1 g ϑΛc1 ,..,Λcg (τ, z) = JCc1 ,..,Ccg (ϑk,a (τ, z) | a ∈ Zg2k ), τ ∈ Hg , z ∈ C(1,g) ,

(29) where (30)

ϑk,a (τ, z) =

X

1

t

t

e2πi 4k (x+a)τ (x+a) +2πi(x+a)z ) .

x∈(2kZ)g

Proof. Let φ be a natural homomorphism from Zn × · · · × Zn to Zn2k × · · · × Zn2k . Let c × ··· × C c be a g-tuple of codewords of C c . Then u∈C 1 g i (31)

1

t

t

e2πiσ( 4k xτ x )+2πi(1,...,1)xz =

X x∈ϕ−1 (u)

X

1

t

e2πiσ( 4k (x+˜u)τ (x+˜u) ))+2πi(1,...,1)(x+˜u)z

t

x∈ϕ−1 (0)

˜ is a preimage of u all of whose entries are non-negative integers less than 2k. Let where u ˜ = (˜ x = (x1 , .., xn )t and u u1 , .., u˜n )t . Then, the equation (31) equals to P

1

xi ∈(2kZ)g

=

t

t t

e2πiσ( 4k (x1 +˜u1 ,..,xn +˜un ) τ (x1 +˜u1 ,..,xn +˜un ))+2πi(1,...,1)(x1 +˜u1 ,..,xn +˜un ) z ϑk,˜u1 (τ, z) · · · ϑk,˜un (τ, z).

The conclusion follows from the fact that the number of a in (Z2k )g which are equal to u˜1 , .., u˜n is exactly Na := na (u). 2

g c , .., C c and Λ c be as in Theorem 4.1. Let M Corollary 4.2 Let C 1 g i c1 ,..,C cg (Xa with a ∈ Z2k ) C be a symmetrized joint weight enumerator in genus g. Then

(32)

ϑΛc1 ,..,Λcg (τ, 0) = MCc1 ,..,Ccg (ϑk,a (τ, 0) | a ∈ Z2k g )

where ϑk,a (τ, z) = ϑk,a (τ, z). Proof. Notice that ϑk,−a (τ, z) = ϑk,a (τ, −z). Especially, ϑk,−a (τ, 0) = ϑk,a (τ, 0). From Theorem 4.1 we obtain the result. 2 Now we show that the theta series defined over the lattices induced from codes are Jacobi forms on some subgroup of Γg . For simplicity we only consider the case when the weight of Jacobi forms(or Siegel modular forms) is integral in this paper. From now on we assume that the length of each code Ci is even. One can obtain a similar result for the case when the length n of the codes is odd. Specifically, Jacobi forms (or Siegel modular forms) of half integral weights are obtained from the complete weight enumerators (or symmetrized weight enumerators) of codes of odd length.

14

Let Γ0 (4) be a subgroup of Γg such that Γ0 (4) = {

A B C D

!

∈ Γg | C ≡ 0

(mod 4)}.

It is known(see [10]) that Γ0 (4) is generated by the following three types of matrices : v(4s) =

Ig 0 4s Ig

! 0

, u(s ) =

Ig s0 0 Ig

!

at 0 0 a−1

and t(a) =

!

,

where s and s0 run over any symmetric g × g integral matrices and a over GL(g, Z). Let χ be a character on Γ0 (4) defined by χ(v(4s)) = χ(u(s0 )) = 1 and χ(t(a)) = det(a−1 ). Theorem 4.3 Let C1 , .., Cg be self-dual codes over Z2k of length n, n ≡ 0 (mod 2), and Λi denote the lattice in Rn induced from Ci . Then the theta series ΘΛ1 ,..,Λg (τ, z) defined as ΘΛ1 ,..,Λg (τ, z) := ϑΛ1 ,..,Λg (2τ, z) is a Jacobi form in J n2 ,nk (Γ0 (4), χ). Moreover, if Ci is a Type II codes for all i then ϑΛ1 ,..,Λg (τ, z) is in J n2 ,nk (Γg ). Proof. To check the first transformation formula of ΘΛ1 ,..,Λg (τ, z), we need only to check it for three types of generators v(s), u(s0 ) and t(a) of Γ0 (4). First, the following functional equation, derived from the Poisson summation formula, (33)

1 τ n −1 t (det ) 2 e2πinkzτ z ϑΛ1 ,..,Λg (τ, z) n i i=1 vol(R /Λi )

ϑΛ1 ,..,Λg (−τ −1 , zτ −1 ) = Qg

implies that, for any symmetric integral matrix s, n

−1 (2s)z t

ϑΛ1 ,..,Λg (τ (2sτ + Ig )−1 , z(2sτ + Ig )−1 ) = (det(2sτ + Ig )) 2 e2πinkz(2sτ +Ig )

ϑΛ1 ,..,Λg (τ, z).

For the unimodularity of Λi , ∀i, implies that vol(Rn /Λi ) = 1. So, with the fact that 2Ig 0 0 Ig

!

v(4s) =

Ig 0 2s Ig

!

2Ig 0 0 Ig

!

,

we check (ΘΛ1 ,..,Λg | n2 ,nk v(4s))(τ, z) = ΘΛ1 ,..,Λg (τ, z). Secondly, from the definition of ϑΛ1 ,..,Λg (τ, z) we obtain that (34) (ΘΛ1 ,..,Λg | n2 ,nk u(s0 ))(τ, z) = ϑΛ1 ,..,Λg (2(τ + s0 ), z) = ϑΛ1 ,..,Λg (2τ, z) = ΘΛ1 ,..,Λg (τ, z). 15

Furthermore, since, for any matrix a ∈ GL(g, Z), ϑΛ1 ,..,Λg (at τ a, za) = ϑΛ1 ,..,Λg (τ, z) and χ(t(a))det(a) = 1, it follows that (ΘΛ1 ,..,Λg | n2 ,nk t(a))(τ, z) = ΘΛ1 ,..,Λg (τ, z). Now, the second transformation formula can be checked directly(see also proof of Theorem3.2 in [2] for more detailed information); using t

t

ϑΛ1 ,..,Λg (τ, z + λτ + µ) = e−2πink(λτ λ +2λz ) ϑΛ1 ,..,Λg (τ, z), ∀λ, µ ∈ Zg , (ΘΛ1 ,..,Λg |nk [λ, µ])(τ, z) = ΘΛ1 ,..,Λg (τ, z). Finally, the condition on a Fourier expansion of ΘΛ1 ,..,Λg (τ, z) follows from the following note(or see proof of Theorem3.2 in [2]); √ √ 4xt x − 2(xt 2k(1, ..1)t )(kn)−1 (xt 2k(1, ..1)t )t = 0. This claims that ΘΛ1 ,..,Λg (τ, z) is in J n2 ,nk (Γ0 (4), χ). Next, we recall that if every code Ci is Type II then Λi are even unimodular for all i and ! Ig s , s integral symthe length n ≡ 0 (mod 8)[1]. Again, since Γg is generated by 0 Ig ! 0 −Ig metric, and , we only need to check the transformation formula for two types Ig 0 ! Ig s of generators of Γg . First, (ϑΛ1 ,..,Λg | n2 ,nk )(τ, z) = ϑΛ1 ,..,Λg (τ + s, z) = ϑΛ1 ,..,Λg (τ, z) 0 Ig since σ(xsxt ) ≡ 0 (mod 2), ∀x ∈ Λ1 × · · · × Λg . Secondly, the Poisson summation formula ng in (33) with i− 2 = 1 implies that (ϑΛ1 ,..,Λg | n2 ,nk

0 −Ig Ig 0

!

)(τ, z) = ϑΛ1 ,..,Λg (τ, z) 0

In this case the character χ on Γg is trivial since χ(u(s )) = χ( each Ci is Type II, then ϑΛ1 ,..,Λg (τ, z) is in J n2 ,nk (Γg ).

0 −Ig Ig 0

!

) = 1. Thus, if 2

Remark 2 1. Note that the character χ is also trivial if the length of each code Ci is divisible by 4. So, if the length of Ci is divisible by 4 then ΘΛ1 ,..,Λg (τ, z) is a Jacobi form in J n2 ,nk (Γ0 (4)). 2. Theorem 4.1 and Theorem 4.3, imply that JC1 ,..,Cg (ϑk,a (2τ, z)|a ∈ Zg2k ) is a Jacobi form of weight n2 and index nk with respect to χ on Γ0 (4) if C1 , .., Cg are self-dual codes over R of length n. 16

Corollary 4.4 Let C1 , .., Cg and Λi be as in Theorem 4.3. Then ΘΛ1 ,..,Λg (τ, 0) is a Siegel modular form of weight n2 with respect to χ on Γ0 (4). Moreover, if all code Ci are Type II, then ϑΛ1 ,..,Λg (τ, 0) is a Siegel modular form on Γg . g

Similarly, we see that MC1 ,..,Cg (ϑk,a (2τ, 0)|a ∈ Z2k ) is a Siegel modular form of weight n2 with respect to χ on Γ0 (4) if C1 , .., Cg are self-dual codes over R of length n. In general, we have the following theorem. Theorem 4.5 Let C1 , .., Cg be self-dual codes over Z2k of length n and Λi denote the lattice in Rn induced from Ci . Then ΘΛc1 ,..,Λcg (τ, z) is a Jacobi form of weight n2 and index nk with respect to χ of Γ where Γ = DI−1 Γ0 (4)DI and DI =

Ig D 0 Ig

!

,

δi D = diag( ), 2

(

δi =

1 if i ∈ I, 0 otherwise.

Proof. First, note that for each a ∈ Rg , ϑk,a (2(τ + D), z) = φI (a)ϑk,a (2τ, z). Here, φI (a) is given in (11). Then the result follows from Theorem 3.7, Theorem 4.1 and Theorem 4.3. 2

Corollary 4.6 Let C1 , .., Cg be self-dual codes over Z2k of length n and Λi denote the lattice in Rn induced from Ci . Then ΘΛc1 ,..,Λcg (τ, 0) is a Siegel modular form of weight n2 with respect to χ on DI−1 Γ0 (4)DI . We remark that only the invariance properties of the weight enumerators have been used to prove the previous few theorems. One can derive a map from an invariant space under the ˜ 2k,g (or, H ˜ 2k,g ) to a ring of Jacobi forms (or, a ring of Siegel modular action of the group G forms).

References [1] E. Bannai, S.T. Dougherty, M. Harada, and M. Oura, “Type II Codes, Even Unimodular Lattices, and Invariant Rings,” IEEE-IT, Vol. 45, No. 4, (1999), 1194-1205. [2] Y. Choie and H. Kim, ”Codes over Z2m and Jacobi forms of genus n”, Jour. of Combinatorial Theory, series A, Vol 95(2), 335 − 348.(2001) [3] Y. Choie and N. Kim, “The complete weight enumerator of Type II code over Z2m and Jacobi forms,” IEEE-IT, Vol 47, No 1, 396 − 399(2001).

17

[4] J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups (2nd ed.), New York: Springer-Verlag, (1993). [5] S.T. Dougherty, “Shadows of Codes”, Yamagata Conference, (October 2000). [6] S.T. Dougherty, T.A. Gulliver and M. Harada, “Type II self-dual codes over finite rings and even unimodular lattices,” J. Alg. Combin., Vol 9, No. 3, (1999), 233-250. (to appear). [7] S.T. Dougherty, M. Harada and M. Oura, “Note on the Biweight Enumerators of SelfDual Codes over Zk ,” submitted. [8] S.T. Dougherty, M. Harada and P. Sol´e, “Shadow lattices and shadow codes,” Discrete Math, Vol. 219, (2000), 49-64. [9] S.T. Dougherty, P. Sol´e, “Shadows of codes and lattices,” submitted to Third Asian Math Conference (2000). [10] T. Ibukiyama, “On Jacobi Forms and Siegel Modular Forms of Half integral Weights,” Comm. Math. Univ. Sancti Pauli, Vol. 41, No.2,(1992), 109-124. [11] F.J. MacWilliams, C. L. Mallows and N.J.A.Sloane, “Generalizations of Gleason’s theorem on weight enumerators of self-dual codes,” IEEE-IT, Vol. 18 (1972), 795–805. [12] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam (1977). [13] J. Wood, “Duality for modules over finite rings and applications to coding theory,” Amer. J. Math., Vol. 121, No. 3, (1999), 555-575. [14] C. Ziegler, “Jacobi forms of Higher Degree,” Abh. Math. Sem. Univ. Hamburg, Vol 59, (1989), 191-224.

18