Codes over , Jacobi forms and Hilbert–Siegel modular forms over

View metadata, citation and similar papers at core.ac.uk European Journal of Combinatorics 26 (2005) 629–650 www.elsevier.com/locate/ejc Codes over F4 , Jacobi forms and Hilbert–Siegel √ modular forms over Q( 5) Koichi Betsumiyaa, YoungJu Choieb a Jobu University, 634-1 Iaesaki, Japan b Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea Received 2 May 2003; accepted 26 April 2004 Available online 19 June 2004 Abstract lattices We study codes over a finite field F4 . We relate self-dual codes over F4 to real 5-modular √ and to self-dual codes over F2 via a Gray map. We construct Jacobi forms over Q( 5) from the complete weight enumerators of self-dual codes over F4 . Furthermore, we relate Hilbert–Siegel forms to the joint weight enumerators of self-dual codes over F4 . © 2004 Elsevier Ltd. All rights reserved. Keywords: Self-dual codes; Integral lattices; Jacobi forms; Hilbert modular forms; Hilbert–Siegel modular forms √ over Q( 5) 1. Introduction Recently there has been intensive research connecting invariant theory and coding theory over fields. The complete weight enumerators of codes over fields can be considered as an invariant polynomial under a certain finite group. It is known that one can construct various modular forms from the weight enumerators of the code by plugging special types of theta-functions [2–5]. One notes that codes over F4 have been studied with respect to their binary image under the Gray map. We construct integral lattices √ induced from codes over F4 and Jacobi forms over the real quadratic field K = Q( 5) and Hilbert modular forms over K from their various weight enumerators. E-mail addresses: [email protected] (K. Betsumiya), [email protected] (Y. Choie). 0195-6698/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2004.04.010 630 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 This paper is organized as follows. In Section 2 the necessary definitions and notations are introduced. In Section 3 the complete weight enumerators of the code over F4 are defined, and MacWilliam’s identities of them are derived. In Section 4, the examples of Type II codes and Type I code over F4 and their complete weight enumerators are studied. In Section 5, we explicitly construct the invariant rings, where the complete weight enumerators of self-dual codes over F4 belong to. The lattice induced by codes over the ring of integers O K is studied in Section 6. In Section 7, by plugging proper Jacobi theta-series to the complete weight enumerators of Type II codes over F4 , we construct √ Jacobi forms over Q( 5). Moreover, we construct an √ algebra homomorphism between a certain invariant ring and that of Jacobi forms over Q( 5). Furthermore, the Hilbert–Siegel modular forms are constructed from the joint weight enumerators of the codes over F4 in Section 8. 2. Notations and definitions Let F4 = F2 (ω) := {0, 1, ω, ω̄ = ω2 } be the finite field of order 4. A code C of length n over F4 is a F4 -subspace of Fn4 . An additive code is a subgroup of (Fn4 , +). An element of C is called a codeword. The inner product on Fn4 is given as [u, v] := n
u jvj, j =1 u = (u j ), v = (v j ) ∈ Fn4 , which is the Euclidean norm. Let C ⊥ = {v ′ | [v ′ , v] = 0 for all v ∈ C}. A code is self-orthogonal if C ⊆ C ⊥ and is self-dual if C = C ⊥ . The Lee composition of a vector x = (x 1 , . . . , x n ) ∈ Fn4 is defined as (n 0 (x), n 1 (x), n 2 (x)), where n 0 (x) is the number of x j = 0, n 2 (x) the number of x j = 1, and n 1 (x) = n − n 0 (x) − n 2 (x). The Lee weight w L (x) of x is defined as n 1 (x) + 2n 2 (x). There is a natural Gray map φ which is a F2 -linear isometry from (Fn4 , Lee distance) onto (F2n 2 , Hamming distance). Here, the Lee distance of two codewords x and y means the Lee weight of x − y; we let, for all x, y ∈ Fn2 , φ(ωx + ω̄y) = (x, y). 3. Weight enumerators We shall now define a series of weight enumerators for codes over F4 . Definition 1. For a code C over F4 define the complete weight enumerator by
 cweC (y0 , y1 , yω , yω2 ) = yana (v), (1) v∈C a∈F4 where n a (v) = |{ j | v j = a}|. Remark 3.1. 1. The complete weight enumerator of a code over F4 is a homogenous polynomial in 4 variables. 2. In this paper, we always use the ordering 0 < 1 < ω < ω2 for the elements in F4 . K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 631 Example 3.2. Let C4 be the [4, 2, 3] linear code over F4 with generator matrix   1 0 ω2 ω . 0 1 ω ω2 The complete weight enumerator φ4 of C4 is φ4 (y0 , y1 , yω , yω2 ) = y04 + y14 + yω4 + yω4 2 + 12y0 y1 yω yω2 . The following was first introduced in [1]. Definition 2. Let C1 , C2 , . . . , C g be codes over F4 . The complete joint weight enumerator for codes C1 , C2 , . . . , C g of length n over F4 is defined as
 na (v1 ,v2 ,...,vg ) g Xa , (2) JC1 ,...,C g (X a | a ∈ F4 ) = (v1 ,v2 ,...,vg )∈C 1 ×C 2 ×···×C g a∈Fg 4 where n a (v1 , . . . , vg ) = |{ j | ((v1 ) j , (v2 ) j , . . . , (vg ) j ) ≡ a}|. 3.1. MacWilliam’s relations Let Tr : F4 → F2 be a trace map and M be a 4 by 4 matrix indexed by the elements of F4 , defined by   1 1 1 1  1 1 −1 −1   M = (M)ν,µ := ((−1)Tr(νµ) ) =  (3)  1 −1 −1 1  . 1 −1 1 −1 Theorem 3.3 (MacWilliam’s Identity for a Complete Weight Enumerator). Let C be a code over F4 . Then cweC ⊥ (yµ | µ ∈ F4 ) = 1 cweC (M · (yµ ) | µ ∈ F4 ). |C| (4) Proof. For each u ∈ Fn4 , consider the map f u : Fn4 → F2 given by f u (v) = Tr(v · u). It is a homomorphism as additive groups. First note that we have |C| if f u (C) = {0} . 0 if f u (C) = F2 u∈C  Now, for cweC ((yµ )) = u∈C ni=1 yu i , u = (u 1 , u 2 , . . . , u i , . . . , u n ), we have   n

 (−1)Tr(vi u i ) yvi  cweC (M · (y − µ)) =
(−1)Tr(v·u) = u∈C i=1 = vi ∈F 4 n

 u∈C v∈Fn4 i=1 (−1)Tr(vi u i ) yvi 632 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 = =
v∈Fn4

(−1) Tr(v·u) u∈C v∈C ⊥ |C| n   n  yv i i=1 yvi = |C|cweC ⊥ ((yµ )). i=1
Theorem 3.4 (MacWilliams’ Identity for a Complete Joint Weight Enumerator). Let C1 , C2 , . . . , C g be codes over F4 and let C̃ denote either C or C ⊥ . Then 1 g JC̃1 ,...,C̃ g (Ya | a ∈ F4 ) = g g i=1 |Ci | where δC̃ = δC̃ i JC1 ,...,C g ((⊗i=1 M δC̃ i g ) · (Ya | a ∈ F4 )), (5) 0 if C̃ = C . 1 if C̃ = C ⊥ Proof. We can prove this using induction on g with Theorem 3.3, so we omit the detailed proof.
4. Type II code over F4 A self-dual code over F4 is said to be Type II if the Lee weight of every codeword is a multiple of 4 and Type I otherwise. In this section we construct some explicit examples of Type II codes over F4 and we give their complete weight enumerators which will be used to study the structure of invariant rings in the following section. We first recall the following properties given in [6]. Proposition 4.1. Let C be a code over F4 . C is Type I (resp. Type II) iff φ(C) is Type I (resp. Type II). Corollary 4.2. There exists a Type II code of length n iff n ≡ 0 (mod 4). 4.1. Examples We define some notations for polynomials as follows:
λi λi λi λi m (λ1 ,λ2 ,λ3 ,λ4 ) = y0 1 y1 2 yω 3 yω24 summed over all distinct permutations (λi1 , λi2 , λi3 , λi4 ) of (λ1 , λ2 , λ3 , λ4 ), m̃ (λ1 ,λ2 ,λ3 ,λ4 ) =
λi λi λi λi y0 1 y1 2 yω 3 yω24 summed over all distinct even permutations (λi1 , λi2 , λi3 , λi4 ) of (λ1 , λ2 , λ3 , λ4 ) and m̂ (λ1 ,λ2 ,λ3 ,λ4 ) = ε((i 1 , i 3 , i 2 , i 4 ))
λi λi λi λi y0 1 y1 2 yω 3 yω24 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 633 summed over all permutations (i 1 , i 3 , i 2 , i 4 ) of (1, 2, 3, 4), where ε is the sign of permutation. m λ is called a monomial symmetric polynomial and m̂ λ is called a skew symmetric polynomial. Here, we give examples of self-dual codes and Type II code and derive their complete weight enumerators which will be referred to in the next section. Example 4.3. Let C4 be the [4, 2, 3] linear code over F4 with generator matrix   1 0 ω2 ω . 0 1 ω ω2 The complete weight enumerator φ4 of C4 is φ4 = cweC4 (Y ) = m (4,0,0,0) + 12m (1,1,1,1) = y04 + y14 + yω4 + yω4 2 + 12y0 y1 yω yω2 . C4 is a Type II code whose minimum Lee weight is 4. Example 4.4.  1 0 0 1  0 0 0 0 Let C8 be the [8, 4, 4] linear code over F4 with generator matrix  0 0 1 1 1 0 0 0 1 1 0 1 . 1 0 1 0 1 1 0 1 0 1 1 1 The complete weight enumerator φ8 of C8 is φ8 = cweC8 (Y ) = m (8,0,0,0) + 14m (4,4,0,0) + 168m (2,2,2,2) = y08 + y18 + yω8 + yω8 2 + 14y04 y14 + 14y04 yω4 + 14y04 yω4 2 + 14y14 yω4 + 14y14 yω4 2 + 14yω4 yω4 2 + 168y02 y12 yω2 yω2 2 . C8 is a Type II code whose minimum Lee weight is 4. Example 4.5.  1 0 0 1  0 0  0 0  0 0 0 0 Let C12 be the [12, 6, 6] linear code over F4 with generator matrix  0 0 0 0 0 ω ω ω2 ω2 1 2 2 ω 1 ω  0 0 0 0 ω 0 ω  1 0 0 0 ω ω2 ω2 0 ω 1  . 0 1 0 0 ω2 ω 0 ω2 1 ω   ω 1 1 1  0 0 1 0 ω2 1 0 0 0 1 1 ω2 1 ω 1 1 The complete weight enumerator φ12 of C12 is φ12 = cweC12 (Y ) = m (12,0,0,0) + 330m (6,2,2,2) + 132m (5,5,1,1) + 165m (4,4,4,0) + 1320m (3,3,3,3) = y012 + y112 + yω12 + yω122 634 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 + 330y06 y12 yω2 yω2 2 + 330y02 y16 yω2 yω2 2 + 330y02 y12 yω6 yω2 2 + 330y02 y12 yω2 yω6 2 + 132y05 y15 yω yω2 + 132y05 y1 yω5 yω2 + 132y05 y1 yω yω5 2 + 132y0 y15 yω5 yω2 + 132y0 y15 yω yω5 2 + 132y0 y1 yω5 yω5 2 + 165y04 y14 yω4 + 165y04 y14 yω4 2 + 165y04 yω4 yω4 2 + 165y14 yω4 yω4 2 + 1320y03 y13 yω3 yω3 2 . C12 is a Type II code whose minimum Lee weight is 8. Example 4.6. Let C20 be the [20, 10, 8] linear code over F4 with generator matrix 1 0 0 1 0 0   0 0  0 0  0 0  0 0  0 0  0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ω ω ω ω 1 1 1 0 ω ω 0 ω ω2 ω2 ω2 1 1 0 ω 1 ω2 1 ω ω2 ω2 0 ω ω 0 ω2 ω 1 0 ω2 ω2 1 ω ω ω 1 1 ω 0 ω2 0 ω2 ω2 ω ω2 ω 0 0 1 ω ω2 ω2 1 ω ω2 ω2 ω2 ω2 ω ω2 0 ω2 1 1 ω2 ω2 ω2 0 ω2 0 1 0 1 1 1 0 1 ω2 ω2 ω2 0 0 ω2 The complete weight enumerator φ20 of C20 is 1  1  ω    ω   2 ω  . 1   1   1   2 ω 0 φ20 = cweC20 = m (20,0,0,0) + 285m (12,4,4,0) + 4560m (11,3,3,3) + 3420m (10,6,2,2) + 380m (9,9,1,1) + 6840m (9,5,5,1) + 855m (8,8,4,0) + 65 835m (8,4,4,4) + 41 040m (7,7,3,3) + 51 300m (6,6,6,2) + 176 472m (5,5,5,5). C20 is a Type II code whose minimum Lee weight is 12. Example 4.7. Let C40 be the [40, 20, 12] linear code over F4 with generator matrix ( I A ), where A is a 20 by 20 matrix constructed by the rows which are shift rotations of (ω, ω, 0, 0, ω, ω, ω, ω2 , 1, 1, ω, ω, ω, 1, ω, 1, 0, 0, 0, 0). The complete weight enumerator of C40 is φ40 = cweC40 (Y ) = m̃ (40,0,0,0) + 50m̃ (28,8,4,0) + 2430m̃ (28,4,4,4) + 6040m̃ (27,7,3,3) + 2320m̃ (26,10,2,2) + 28 600m̃ (26,6,6,2) + 160m̃ (25,13,1,1) + 17 240m̃ (25,9,5,1) + 446 424m̃ (25,5,5,5) + 1710m̃ (24,12,4,0) + 13 215m̃ (24,8,8,0) + 827 980m̃ (24,8,4,4) + 319 000m̃ (23,11,3,3) + 3011 000m̃ (23,7,7,3) + 28 520m̃ (22,14,2,2) + 2015 920m̃ (22,10,6,2) + 28 299 120m̃ (22,6,6,6) + 1753 160m̃ (21,9,9,1) + 44 465 880m̃ (21,9,5,5) + 600m̃ (21,17,1,1) + 314 560m̃ (21,13,5,1) + 2m̃ (20,20,0,0) + 9450m̃ (20,16,4,0) + 258 630m̃ (20,12,8,0) + 17 683 670m̃ (20,12,4,4) + 125 056 940m̃ (20,8,8,4) K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 635 + 2070 160m̃ (19,15,3,3) + 80 902 440m̃ (19,11,7,3) + 762 302 520m̃ (19,7,7,7) + 76 160m̃ (18,18,2,2) + 14 765 640m̃ (18,14,6,2) + 985 911 040m̃ (18,10,6,6) + 70 425 440m̃ (18,10,10,2) + 777 160m̃ (17,17,5,1) + 14 741 920m̃ (17,13,9,1) + 372 158 960m̃ (17,13,5,5) + 2112 617 800m̃ (17,9,9,5) + 671 115m̃ (16,16,8,0) + 47 132 930m̃ (16,16,4,4) + 1224 223 930m̃ (16,12,8,4) + 2472 640m̃ (16,12,12,0) + 8657 100 585m̃ (16,8,8,8) + 229 717 840m̃ (15,15,7,3) + 8954 480 400m̃ (15,11,7,7) + 949 581 600m̃ (15,11,11,3) + 3013 467 680m̃ (14,14,6,6) + 215 304 040m̃ (14,14,10,2) + 14 364 714 760m̃ (14,10,10,6) + 7031 906 800m̃ (13,13,9,5) + 49 235 520m̃ (13,13,13,1) + 39 902 290 560m̃ (13,9,9,9) + 31 831 334 340m̃ (12,12,8,8) + 4501 219 800m̃ (12,12,12,4) + 37 039 712 880m̃ (11,11,11,7) + 68 470 672 704m̃ (10,10,10,10). C40 is a Type II code. Example 4.8 (Type I Codes). Let C2 be a [2, 1, 2] linear code over F4 with generator matrix (1 1). Then C2 is a Type I code with the complete weight enumerator φ2 = cweC2 (Y ) = m (2,0,0,0). Let C6 be the [6, 3, 3] linear code over F4 with generator matrix   1 1 1 1 1 1  0 0 0 1 ω ω2  . 1 ω ω2 0 0 0 Then C6 is a Type I code with the complete weight enumerator φ6 = cweC6 (Y ) = m (6,0,0,0) + 6m (3,1,1,1) + 9m (2,2,2,0). Let C16 be the [16, 8, 4] linear code over F4 with generator matrix 1 0 0 0 0 0 0 0 ω 1 ω 1 ω 1 ω2 0  0  0  0  0  0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 ω 1 ω 1 ω2 1 ω 1 ω 1 ω2 1 ω 1 ω 1 ω2 1 ω 1 ω 1 ω2 1 ω 1 ω 1 ω2 1 ω 1 ω 1 ω2 1 ω 1 ω 1 ω Then C16 is the Type I code with the complete weight enumerator φ16 = cweC16 (Y ) = m̃ (16,0,0,0) 1 ω 1 ω 1 ω 1 1  ω   1   ω  . 1   ω   1  ω2 636 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 + 12m̃ (12,4,0,0) + 48m̃ (10,2,2,2) + 38m̃ (8,8,0,0) + 192m̃ (8,6,2,0) + 84m̃ (8,4,4,0) + 384m̃ (8,4,2,2) + 512m̃ (7,5,3,1) + 64m̃ (6,6,4,0) + 1312m̃ (6,6,2,2) + 1344m̃ (6,4,4,2) + 3072m̃ (5,5,3,3) + 6936m̃ (4,4,4,4). 5. Invariant rings arising from codes In this section we study the invariant rings where the complete weight enumerators of Type II codes over F4 belong to. Let M ∗ , g1 , g2 , g3 and g4 be matrices indexed by F4 , where M ∗ := 21 M, (M)uv = (−1)Tr(uv), 1 for v + 1 = u (g1 )uv = 0 otherwise, (g3 )uv = i wt B (u) 0 for v = u otherwise, (g2)uv = 1 0 for ωv = u otherwise, (g4 )uv = and Here, wt B (u) = ι(Tr(ωu)) + ι(Tr(ω2 u)), with a map ι ι(0) = 0. Explicitly, these matrices are    0 1 1 1 1 1  1 1 1 −1 −1 ∗ , M =  g1 =  0 2  1 −1 −1 1  0 1 −1 1 −1    1 0 0 0 1 0 0 0 0 0 1  0 −1 0  g3 =  g2 =  0 1 0 0, 0 0 i 0 0 0 0 0 1 0   1 0 0 0 0 1 0 0  and g4 =  0 0 0 1. 0 0 1 0 1 0 for v 2 = u otherwise. : F2 → Z, defined by ι(1) = 1, 1 0 0 0 0 0 0 1  0 0 , 0 i  0 0 , 1 0 Proposition 5.1. Let G be the group generated by {M ∗ , g1 , g2 , g3 }. Then G = M ∗ , T1 , T2 , T3 , where 1 0 T1 :=  0 0  0 0 −1 0 0 −i 0 0  0 0  , 0  −i 1 0 T2 :=  0 0  0 −i 0 0 0 0 −i 0  0 0  , 0  −1 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 1 0 T3 :=  0  0 0 i 0 0 0 0 1 0 637 0 0  . 0  −i  Proof. The relation G ⊃ M ∗ , T1 , T2 , T3 is immediate from the fact that T1 = g33 , T2 = g22 g33 g2 and T3 = g32 g2 g3 g22 . Moreover, the order of each group is 3840. This completes the proof.
5.1. Self-dual codes over F4 First, we recall the following two known results concerning invariant rings for the selfdual codes over F4 . We denote the conjugate code of C over F4 by C̄ = {(c12, c22 , . . . , cn2 ) | (c1 , c2 , . . . , cn ) ∈ C}. Theorem 5.2 (Invariant Ring for Self-Dual Codes with cweC = cweC̄ (see [8, 9])). Let G ′SD be the group generated by {M ∗ , g1 , g2 , g4 }. Then the order of G ′SD is 384 and the Molien series of G ′SG is 1 . (1 − t 2 )(1 − t 4 )(1 − t 6 )(1 − t 8 ) (6) The complete weight enumerator of any self-dual code C over F4 with cweC = cweC̄ ′ ′ belongs to C[Y ]G SD . Moreover, C[Y ]G SD = C[φ2 , φ4 , φ6 , φ8 ]. Theorem 5.3 (Invariant Ring for Self-Dual Codes (see [8, 9])). Let G SD be the group generated by {M ∗ , g1 , g2 }. Then the order of G SD is 192 and the Molien series of G SD is t 16 + 1 . (1 − t 2 )(1 − t 4 )(1 − t 6 )(1 − t 8 ) (7) The complete weight enumerator of every self-dual code over F4 belongs to C[Y ]G SD . Moreover, C[Y ]G SD = R ⊕ ψ16 R, where R = C[φ2 , φ4 , φ6 , φ8 ] and ψ16 = m̂ (10,4,2,0) − 3m̂ (8,6,2,0) − 8m̂ (7,5,3,1). Using the above two theorems concerning self-dual codes over F4 we get the following proposition. Proposition 5.4. The minimum length of any self-dual code C over F4 with cweC = cweC̄ is 16. For any such code C, cweC (Y ) − cweC̄ (Y ) = kψ16 for some k ∈ Z. Here, ψ is given in (8). Proof. First, we note C16 = C̄16 and 64ψ16 = φ̄16 − φ16 . ′ Since cweC = cweC̄ , cweC (Y ) ∈ C[Y ]G SD \C[Y ]G SG . Let P16 = {cweC (Y ) − cweC̄ (Y ) | C = C̄, Length(C) = 16}. (6) and (7) imply that dimC ( P16 ) = 1. Moreover, the coefficients of any element in P16 are integers. So, the result follows.
638 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 Finally, we derive the following conclusion about the minimal weight of the self-dual code over F4 . Corollary 5.5. For a self-dual code C of length 16 over F4 with cweC = cweC̄ , both the minimum Hamming weight and the minimum Lee weight are at most 6. Proof. According to ψ16 , the complete weight enumerator of any self-dual [16, 8] code C = C̄ has the monomial y010 yω4 yω2 2 . It corresponds to codewords of Hamming weight 6 and of Lee weight 6.
5.2. Type II codes over F4 Now we study the structure of invariant rings where the complete weight enumerators of Type II codes over F4 belong to. We state one of our main results. Theorem 5.6 (Invariant Ring for Type II Codes with cweC = cweC̄ ). Consider the group G ′ = M, g1 , g2 , g3 , g4 with order 7680. Then 1. The Molien series of G ′ is (t 4 − 1)(t 8 1 . − 1)(t 12 − 1)(t 20 − 1) Note that G ′ is the unitary reflection group No. 29 in the Sephard–Todd list. 2. The invariant ring of G ′ is ′ C[y0 , y1 , yω , yω2 ]G = C[φ4 , φ8 , φ12 , φ20 ] as a polynomial ring. Before we prove the theorem, we note the following. Proposition 5.7. Let φi , i = 4, 8, 12, 20, be the complete weight enumerators of the Type II code given before. Then φ4 , φ8 , φ12 and φ20 are algebraically independent. Proof. First, we introduce an order among monomials by the lexicographic order of exponents as ′ ′ ′ ′ y0a y1b yωc yωd 2 < y0a y1b yωc yωd 2 if a < a ′ or a = a ′ and b < b ′ or a = a ′ , b = b′ and c < c′ or a = a ′ , b = b′ , c = c′ and d < d ′ . Let f be a homogenous polynomial. We define In( f ) by the minimal monomial of f with respect to the order. It is easy to see that In( f g) = In( f )In(g). We denote φ8′ = φ8 − φ42 , ′ φ12 = φ12 + 14 φ4 φ8 − 54 φ43 ′ φ20 = φ20 + 179 5 336 φ4 + 265 3 1008 φ4 φ8 − 100 2 63 φ4 φ12 − 5 2 72 φ4 φ8 − 5 36 φ8 φ12 . ′ ) = y 4 y 4 y 4 and In(φ ′ ) = Then In(φ4 ) = yω4 2 , In(φ8′ ) = yω4 yω4 2 , In(φ12 1 ω ω2 20 8 12 20 17 4 y0 y1 yω yω2 . We assume g(x 1 , x 2 , x 3 , x 4 ) is a non-trivial homogenous polynomial in 639 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 ′ ′ ′ ′ ′ , φ ′ ) = 0. Let x a x b x c x d and x a x b x c x d C[x 1 , x 2 , x 3 , x 4 ] such that g(φ4 , φ8′ , φ12 1 2 3 4 20 1 2 3 4 ′ ′ ′ ′ be monomials of g(x 1 , x 2 , x 3 , x 4 ) such that x 1a x 2b x 3c x 4d > x 1a x 2b x 3c x 4d . Then ′ ′ ′ ′ ′c φ ′d ) < In(φ a φ ′b φ ′c φ ′d ). If we let x a x b x c x d be the maximal monomial In(φ4a φ8′b φ12 4 8 12 20 1 2 3 4 20 ′c ′d ′ , φ ′ ). This is a of g, the monomial In(φ4a φ8′b φ12 φ20 ) cannot be cancelled in g(φ4 , φ8′ , φ12 20 contradiction.
′ Proof of Theorem 5.6. It is immediate that C[y0, y1 , yω , yω2 ]G ⊃ C[φ4 , φ8 , φ12 , φ20 ]. By Proposition 5.7, it is shown that the dimensions of the homogenous components ′ coincide. Therefore, C[y0, y1 , yω , yω2 ]G = C[φ4 , φ8 , φ12 , φ20 ].
Finally, we state our main theorem regarding the invariant ring for Type II codes. Theorem 5.8 (Invariant Ring for Type II Codes). Consider the group G = M ∗ , g1 , g2 , g3 with order 3840. Then 1. The Molien series of G is (t 4 − 1)(t 8 t 40 + 1 . − 1)(t 12 − 1)(t 20 − 1) 2. The invariant ring C[y0, y1 , yω , yω2 ]G = R ψ40 R, where R := C[φ4 , φ8 , φ12 , φ20 ] and ψ40 = m̂ (28,8,4,0) − 32m̂ (25,9,5,1) − 5m̂ (24,12,4,0) + 272m̂ (22,10,6,2) + 64m̂ (21,13,5,1) + 10m̂ (20,16,4,0) − 125m̂ (20,12,8,0) − 768m̂ (19,11,7,3) − 304m̂ (18,14,6,2) + 608m̂ (17,13,9,1) + 969m̂ (16,12,8,4). The following proposition tells about the minimal length of a certain family of Type II codes and about relations between the complete weight enumerators of C and its conjugate C̄. Proposition 5.9. The minimum length for Type II codes C over F4 with wecC = cweC̄ is 40. For any such code C, cweC (Y ) − cweC̄ (Y ) = kψ40 for some k ∈ Z. Proof. Note that 40ψ40 = φ̄40 − φ40 . Since the conclusion can be derived in a similar manner to the proof of Proposition 5.4, we omit the detailed proof.
As a corollary, we can also make a statement about the upper bound of the minimal weight of a certain family of Type II codes over F4 . Corollary 5.10. For any Type II code C of length 40 over F4 , both the minimum Hamming weight and the minimum Lee weight are at most 12. 640 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 Proof. From the definition of ψ40 given in (8), the complete weight enumerator of any Type II [40, 20] code C with cweC = cweC̄ has the monomial y028 yω8 yω4 2 that corresponds to codewords of Hamming weight 12 and of Lee weight 12.
6. Lattices From now on we let K = Q(α), α = (1 + known to be √ 5)/2. The ring of integers O K of K is O K = Z[α] = Z + αZ. For each β ∈ K , β (2) denotes the algebraic conjugate of β = β (1). Then the trace map Tr K /Q : K → Q is defined as Tr K /Q (x) = x (1) + x (2) . Note that Tr K /Q : O K → Z. A lattice Λ in K n can be considered as a free O K -module. The standard inner product is attached:
v·u = Tr K /Q (vi u i ), ∀v = (v j ), u = (u j ) ∈ Λ. (8) Define the dual lattice Λ∗ = {u ∈ K n | u · v ∈ Z for all v ∈ Λ}. A lattice is integral if Λ ⊆ Λ∗ . The norm of a vector v is given by N(v) = v · v. If Λ is a unimodular lattice and N(v) ∈ 2Z for all v ∈ Λ, then Λ is said to be an even lattice. To construct the lattice from a code C over F4 we recall the following elementary property. Proposition 6.1. Let F be a real quadratic field with the discriminant dF = m. Then the ideal (2) in its ring of integers OF is decomposed as follows: (2) = PP ′ , P = P ′ , N(P) = N(P ′ ) = 2 if m ≡ 1 (mod 8) . (2) = P, N(P) = 22 if m ≡ 5 (mod 8) Here, P, P ′ are distinct prime ideals in O K with norm N(P), N(P ′ ), respectively. Remark 6.2. According to the √ above proposition, since (2) = P, we have N(P) = |O K /(2)| = 4, where K = Q( 5). Now, consider the following reduction map h modulo the ideal (2): h : O K → F4 , (9) defined by h(a + bα) = a (mod 2) + b (mod 2)ω, ∀a + bα ∈ O K . Note that this map h is a ring homomorphism. Then the following naturally induced homomorphism for a code over F4 is given by h̃ : OnK → Fn4 . (10) It is easy to check that h̃ −1 (C), the preimage of a code C defined over F4 , is a free O K module. So, we define the lattice induced from a code C as follows: 1 1 Λ(C) := √ h̃ −1 (C) = √ {v ∈ OnK | v (mod 2O K ) ∈ C}. 2 2 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 641 Theorem 6.3. If C is a self-dual code over F4 , then Λ(C) is an integral lattice. Moreover, if C is Type II, then Λ(C) is an even lattice. Proof. If v = (v j = a j + b j ω) j and v ′ = (v ′j = c j + d j ω) j are vectors in C, then v j v ′j = 0. Consider a single coordinate: (a j + b j ω)(a ′j + b ′j ω) = (a j a ′j + b j b′j ) + (b j a ′j + a j b′j + b j b′j )ω = 0. √ √ √ √ Now (1/ 2)h̃ −1 (v) = (1/ 2)(a j + b j α + (2)O K ) j and (1/ 2)h̃ −1 (v ′ ) = (1/ 2)(a ′j + b′j α + 2O K ) j . We need to show that the vector z in h̃ −1 (v) has an integral inner √ product with any vector z ′ in (1/ 2)h̃ −1 (v ′ ), i.e., Tr K /Q (z j z ′j ) ∈ Z. Consider a single coordinate: Note that (a j + b j ω)(a ′j + b ′j ω) = (a j a ′j + b j b′j ) + (b j a ′j + a j b′j + b j b′j )ω = 0 if and only if (a j a ′j + b j b′j ≡ 0 (mod 2), b j a ′j + a j b′j + b j b′j ≡ 0 (mod 2). So, we have 1 1 1 √ h̃ −1 (a j + b j ω) √ h̃ −1 (a ′j + b ′j ω) = Tr K /Q ((a j a ′j + b j b′j ) + (b j a ′j + a j b′j 2 2 2 + b j b′j )α + (2)O K ) = 1 1 Tr K /Q (a j a ′j + b j b′j ) + Tr K /Q ((b j a ′j 2 2 + a j b′j + b j b′j )α) ∈ Z. This means that Λ(C) is an integral lattice. This together with the above computation shows that the image of a Type II code is an even lattice.
7. Constructions of Jacobi forms over K There has been intensive research connecting invariant theory and coding theory. Specifically, the complete weight enumerator of codes, seen as an invariant polynomial under a certain finite group, is used to construct various modular forms plotting special types of theta-functions [2]. In this section we extend this idea by studying the connections between the complete weight enumerators of codes over F4 and Jacobi forms over K . More precisely, the Jacobi theta-series formed from the complete weight enumerators of the codes over F4 is a Jacobi form over the real quadratic field K . Also, Hilbert–Siegel modular forms of higher genus have been derived from the joint weight enumerators of codes over F4 . 7.1. Jacobi group and Jacobi forms over the real quadratic field K We recall the definition of Jacobi forms over K and theta-series. We follow the definitions given in [10]. 642 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 The Jacobi group of K will be denoted by Γ J (K ) := S L 2 (O K ) ∝ O2K . This group acts on H2 × C2 , where H denotes the complex upper half plane. Variables J of this space will be listed  as ( , ) := (τ1 , τ2 , z 1 , z 2 ). The action of Γ (K ) on the space H2 × C2 are given by  α γ β δ  α γ · ( , ) := β δ  ∈ S L 2 (O K ), z1 z2 α (1) τ1 + β (1) α (2) τ2 + β (2) , (2) , (1) , (2) (1) (1) (2) (1) γ τ1 + δ γ τ2 + δ γ τ1 + β γ τ2 + β (2)  and, for all [λ, µ] ∈ O2K , [λ, µ] · ( , ) := (τ1 , τ2 , z 1 + λ(1) τ1 + µ(1) , z 2 + λ(2) τ2 + µ(2) ). Remark 7.1. It is known (see [7]) that S L 2 (O K ) is generated by the matrices     1 β 0 −1 , ∀β ∈ O K . , 0 1 1 0 We first introduce the following notation for convenience: for τ ∈ H2 , γ , δ ∈ O K , denote 2  (γ ( j ) τ j + δ ( j )), N (γ τ + δ) := j =1 2 e 2πi Tr K /Q (m cτcz+d ) e−2πi Tr K /Q (m(λ := 2  e 2πim ( j ) c( j ) z 2j c( j ) τ j +d ( j ) , j =1 2 τ +2λz)) := 2  e−2πim ( j ) (λ( j )2 τ j +2λ ( j )z j) . j =1 Definition 3. Given k ∈ (1/2)Z and m ∈ O K , a function f : H2 × C2 → C is said to be a Jacobi forms of weight k and index m for the real quadratic field K if it is an analytic function satisfying cz 2 1. ( f |k,m M)( , ) := N (cτ + d)−k e−2πi Tr K /Q (m cτ +d ) f (M · ( , ))   ∗ ∗ = f ( , ), ∀M = ∈ S L 2 (O K ). c d 2. ( f |m [λ, µ])( , ) := e−2πi Tr K /Q (m(λ τ +2λz j ) f ( , [λ, µ] · ) ∀[λ, µ] ∈ O2K . = f ( , ), 2 And it has the following Fourier expansion:
3. f( , )= c(α, β)e2πi Tr K /Q (ατ +βz). α,β∈δ −1 K ,α≥0 Here δ −1 K is the inverse different of K . K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 643 Remark 7.2. 1. The C-vector space of Jacobi forms of weight k and index m for the field K is denoted by Jk,m (Γ1 (O K )).  from a Jacobi 2. Note that letting = 0 one obtains a Hilbert modular form f ( , 0) form. 7.2. Theta-series The following theta-function was first introduced and studied in [10] to show the correspondence between the space of Jacobi forms over the real quadratic field and the space of the vector valued modular forms. Let µ ∈ O K be the corresponding preimage of µ̃ ∈ F4 under the reduction map h given before, so, µ ∈ S := {0, 1, α, α 2 = α + 1} (sometimes we identify the set S as {0, 1, ω, ω + 1}.) Now, for each µ, consider the following theta-series:
r2 τ (11) e2πi Tr K /Q ( 4 +r z) . θ1,µ ( , ) := r∈δr−1 K ,r≡µ (mod(2)) Then, by the Poisson summation formula, the theta-series satisfies the following transformation formula [10]. Lemma 7.3.   1 1. θ1,µ | 1 ,1 2 0   0 2. θ1,µ | 1 ,1 2 1 β 1  µ2 β ( , ) = e2πi Tr K /Q ( 4 ) θ1,µ ( , ), ∀β ∈ O K .   0 −1  χ 1 0 −1 ( , )= 0 2
µν e2πi Tr K /Q ( 4 ) θ1,ν ( , ). × ν∈O K /2O K Here, χ  0 1 −1 0 4 = 1. Remark 7.4. The transformation formula of the theta-series given in Lemma 7.3 implies that, for any β = a + bα ∈ O K , a, b ∈ Z, (θ1,0 ( + β, ), θ1,1 ( + β,  T1 (θ1,0 ( , ), θ1,1 ( ,     T2 (θ1,0 ( , ), θ1,1 ( , =  I (θ1,0 ( , ), θ1,1 ( ,    T3 (θ1,0 ( , ), θ1,1 ( , ), θ1,α ( + β, ), θ1,α 2 ( + β, ))t ), θ1,α ( , ), θ1,α ( , ), θ1,α ( , ), θ1,α ( , ), θ1,α 2 ( , ), θ1,α 2 ( , ), θ1,α 2 ( , ), θ1,α 2 ( ,  ))t , if a ≡ 1, b ≡ 0 (mod 2)   ))t , if a ≡ 1, b ≡ 1 (mod 2) . ))t , if a ≡ 0, b ≡ 0 (mod 2)     ))t , if a ≡ 0, b ≡ 1 (mod 2) Here, Ti are matrices in Proposition 5.1 and I is an identity matrix. 7.3. The complete weight enumerators of the codes over F4 and Jacobi forms over K In this section we construct a theta-series defined over the lattice induced from codes C over F4 . Furthermore, we show that the constructed theta-series is a Jacobi form over K if the codes are Type II. 644 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 For each Y in the lattice Λ, consider the theta-series ΘΛ,Y : H2 × C2 → C associated with a lattice:
x·x ΘΛ,Y ( , ) := e2πi Tr K /Q ( 2 τ +x·Y z). (12) x∈Λ The following theorem gives a connection between a theta-series defined over the lattices induced from codes and their complete weight enumerators. Theorem 7.5. Let √C be a code over F4 . Let Λ(C) be a lattice induced from C over F4 , i.e., Λ(C) = (1/ 2)h̃ −1 (C). The following theta-series ΘΛ(C),√2(1,...,1) ( , ) associated with Λ(C) is exactly the same as the complete weight enumerator cweC (y0 , y1 , yω , yω2 ) evaluated at θ1,µ ( , ), µ ∈ S. In other words, ΘΛ(C),√2(1,...,1) ( , ) = cweC (θ1,µ ( , ) | µ ∈ S), (13) where {θ1,µ } is given in Lemma 7.3. √ √ Proof. Note that 2 = (1/ 2)(2, . . . , 2, 2) ∈ Λ(C). Let v = (v1 , . . . , vn ) be any given codeword in C and, for each µ ∈ F4 , let n µ (v) = |{ j | v j = µ}|. If we let h̃(ṽ) = v, then the image can be arranged in the following form: {h̃ −1 (v)} = {h̃ −1 (0)+ ṽ | ṽ = (ṽ j ), ṽ j = a j + b j w, 0 ≤ a j , b j < 2} and the number of µ in (ṽ1 , . . . , ṽn ) is exactly n µ (v). Thus, for each v ∈ C,
e2πi Tr K /Q ( x·x 2 τ +x· √ 2z) x∈ √1 h̃ −1 (v) 2 = x 1 ∈2O K +ṽ1  = =

··· e
e2πi Tr K /Q ( x2 2πi Tr K /Q ( 41 τ +x 1 z 1 ) θ1,µ ( , )nµ (v).
e2πi Tr K /Q ( (x+ṽ)·(x+ṽ) τ +(x+ṽ)·1z) 4 x∈h̃ −1 (0) x n ∈2O K +ṽn x 1 ∈2O K +ṽ1  =  (x 2 +x 2 +···+x n2 ) 1 2 τ +x 1 z+···+x n z) 4 ··· 
e 2πi Tr K /Q ( x n ∈2O K +ṽn x n2 4 τ +x n z)  µ∈S Therefore, we have ΘΛ(C),√2 ( , ) = cweC (θ1,µ ( , ) | µ ∈ S). This finishes the proof.
Theorem 7.6. Let C be a Type II code of length n over F4 . Then cweC (θ1,µ ( , ) | µ ∈ F4 ) is a Jacobi form of weight n/2 and index n over K . Proof. For convenience, let g( , ) := cweC (θ1,µ ( , ) | µ ∈ S). To check the modularity of g( , ), it is enough to  check the transformation formula under the two types of   1 β 0 −1 generators (see Remark 7.1) 0 1 and 1 0 of Γ1 (O K ), ∀β ∈ O K . K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 645 First, for any β = a + bα ∈ O K , a, b ∈ Z,    1 β ( , ) = cweC (θ1,µ ( + β, ) | µ ∈ S) g | n2 ,n 0 1 = cweC (e2πi Tr K /Q ( µ2 β 4 ) θ1,µ ( , ) | µ ∈ F4 ) (from Lemma 7.3)   cweC (T1 · (θ1,µ ( , ))) if a ≡ 1, b ≡ 0 (mod 2)      cweC (T2 · (θ1,µ ( , ))) if a ≡ 1, b ≡ 1 (mod 2) = cweC (I · (θ1,µ ( , ))) if a ≡ 0, b ≡ 0 (mod 2)       cweC (T3 · (θ1,µ ( , ))) if a ≡ 0, b ≡ 1 (mod 2) = g( , ) (from Remark 7.4). Secondly,   0 n g | 2 ,n 1  nz 2 n −1 ( , ) = N (τ )− 2 e−2πi Tr K /Q ( τ ) 0      −1 −1 z 1 z 2  µ∈S , , , × cweC θ1,µ τ1 τ2 τ1 τ2      1 2 N (τ ) 2 0 −1 − n2 −2πi Tr K /Q ( nzτ ) = N (τ ) e cweC χ −1 0 22  z2 × e2πi Tr K /Q ( τ ) M · (θ1,µ ( , ) | µ ∈ S) 1 cweC (M · (θ1,µ ( , ) | µ ∈ S)) 2n (since C is Type II) = cweC ⊥ ((θ1,µ | µ ∈ S)) = = cweC (θ1,µ ( , ) | µ ∈ S) = g( , ). Next, one needs to check the elliptic property; for any λ, µ ∈ O K , 2 e2πi Tr K /Q (n(λ τ +λz)) g( , + λ + µ) √ √
√ x·x+2nλ2 = e2πi Tr K /Q ( 2 τ +(x+ 2λ)· 2z) (since λ ∈ O K , replace x + 2λ → x) x∈Λ(C) = g( , ). Finally, the proper Fourier expansion can be checked easily and we omit the detailed proof.
7.4. Invariant space and space of Jacobi forms over K In the previous section, we showed how to construct Jacobi forms over K from the complete weight enumerators of the Type II codes C over F4 . This comes from the invariant property of the complete weight enumerator of C over F4 . More generally, we construct an 646 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 algebra homomorphism from the algebra of the invariant polynomials, where the complete weight enumerators of the Type II codes belong to, to the algebra of Jacobi forms over K . Let G be a matrix group and let C[X] be a polynomial ring. Then G acts on C[X] in the following way: g · F(X) := F(g · X), ∀g ∈ G, ∀F ∈ C[X]. Then we define that C[X]G := {F ∈ C[X] | g · F(X) = F(g · X), ∀g ∈ G}. Theorem 7.7. Let C[y0 , y1 , yω , yω2 ]G := { f (y0 , y1 , yω , yω2 ) ∈ C[y0 , y1 , yω , yω2 ] | g · f (y0 , y1 , yω , yω2 ) = f (y0 , y1 , yω , yω2 )}. Then the following map Φ : C[y0 , y1 , yω , yω2 ]G → ⊕ℓ,m Jℓ,m (Γ1 (O K )), given by Φ(F(y0 , y1 , yω , yω2 )) = F(θ1,0 , θ1,1 , θ1,α , θ1,α 2 ), ∀F ∈ C[y0, y1 , yω , yω2 ]G is an algebra homomorphism. Remark 7.8. Note that C[y0 , y1 , yω , yω2 ]G ⊂ ⊕ℓ>0 C[y0, y1 , yω , yω2 ]4ℓ := {F ∈ C[y0 , y1 , yω , yω2 ] | degree(F) ≡ 0 (mod 4)}. So, the image Φ in Theorem 7.7 belongs to the space of Jacobi forms with even integral weight. Proof. It is enough to check the transformation formula for g( , ) := F(θ1,0 ( , ), θ1,1 ( , ), θ1,α ( , ), θ1,α 2 ( , )); with degree(F) = ℓ, ∀β ∈ O K ,    1 β · ( , ) = F((θ1,µ ( + β, ) | µ ∈ S)) = F(Ti (θ1,µ ( , ) | µ ∈ S), g 0 1 for some Ti ∈ {T1 , T2 , T3 } because of the transformation formula of θ1,µ ( , ). Next,        1 1 z1 z2 1 z 1 z 2  1 g − ,− , , = F θ1,µ − , − , , µ ∈ S τ1 τ2 τ1 τ2 τ1 τ2 τ1 τ2       1 τ 2 2πi Tr K /Q (ℓ z2 ) 0 −1 τ e =F χ N 1 0 2  × M · (θ1,µ ( , ) | µ ∈ S) ℓ z2 = N (τ ) 2 e2πi Tr K /Q (ℓ τ ) F(M ∗ (θ1,µ ( , ) | µ ∈ S)t ) (since ℓ = degree(F) ≡ 0 (mod 4)) ℓ z2 = N (τ ) 2 e2πi Tr K /Q (ℓ τ ) F((θ1,µ ( , ) | µ ∈ S)t ). Since the condition at the cusps and the elliptic property can also be checked in a similar manner as in the proof of Theorem 7.6, we omit the detailed proof.
647 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 7.5. Explicit example Note that one can construct an elliptic modular form from Jacobi Hilbert forms or from Hilbert forms by specializing variables, namely, τ1 = τ2 , z 1 = z 2 = 0. In this section we derive some explicit relations among the various theta-functions using the map Φ studied in the previous section. √ For K = Q( 5), the theta-series defined in (11) are θ1,0 (τ, τ, 0, 0) =
q 2a 2 +3b2 +2ab , a,b∈Z 1 θ1,1 (τ, τ, 0, 0) = q 2
2 +3b2 +2ab+2a+b , a,b∈Z 3 θ1,α (τ, τ, 0, 0) = q 4 q 2a
q 2a 2 +3b2 +2ab+a+3b , a,b∈Z 7 θ1,α+1 (τ, τ, 0, 0) = q 4
q 2a 2 +3b2 +2ab+2a+4b . a,b∈Z So, φ4 (θ1,0 (τ, τ, 0, 0), θ1,1 (τ, τ, 0, 0), θ1,α (τ, τ, 0, 0), θ1,α+1 (τ, τ, 0, 0)) 4  4 

2a 2 +3b2 +2ab+2a+b− 21 2a 2 +3b2 +2ab q + q = a,b∈Z a,b∈Z + 
a,b∈Z + 12 × q 2a 2 +3b2 +2ab+a+3b+ 43  
q 2a 2 +3b2 +2ab q
+ q 
q 2a 2 +3b2 +2ab+2a+4b+ 43 a,b∈Z 2a 2 +3b2 +2ab+2a+b− 21 a,b∈Z a,b∈Z
 4 2a 2 +3b2 +2ab+a+3b+ 43 a,b∈Z 
q 4  2a 2 +3b2 +2ab+2a+4b+ 43 a,b∈Z  = E 4 (τ ). Here, E 4 (τ ) is the elliptic Eisenstein series of weight 4 defined by

E 4 (τ ) = 1 + 240 σ3 (n)q n , σ3 (n) = d 3. n>0 0<d|n 8. Construction of Hilbert–Siegel modular form of genus g over K In this section, we consider a higher genus Hilbert–Siegel modular form over K and derive a connection with the joint weight enumerators of codes over F4 and the symmetrized weight enumerators of codes as well. 648 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 8.1. Hilbert–Siegel modular form of genus g over K Let Hg := {τ ∈ Mg×g (C) | Im(τ ) > 0} be the Siegel upper half plane and let Γg (O K ) := S P2g (O K ) be the symplectic group acting on Hg by  A C B D  · τ := (Aτ + B)(Cτ + D)−1 . We define the Hilbert–Siegel modular form of genus g over K . Definition 4. A holomorphic function F : H2g → C is called a Hilbert modular form of weight k and genus g over K if   ∗ ∗ ∈ Γg (O K ), F(M · τ ) = N (Det(Cτ + D))−k F(τ ), ∀M = C D with a proper holomorphic condition at each cusp in the case of g = 1. (g) Consider, for each µ ∈ S g = S × S × · · · × S, the theta-series θ1,µ : H2g → C, (g) θ1,µ (τ ) =
e2πi Tr K /Q ( rτ r t 4 ) . (14) g g r∈(δ −1 K ) ,r≡µ (mod(2O K ) ) Then the following can be derived. (g) Lemma 8.1. Let θ1,µ (τ ) be the function defined in (14). Then it satisfies the following transformation formula:  g 0 −1 χ 1
−1 0 (g) θ1,µ (−τ −1 ) = (−1)Tr(µ·ν) θ1,ν (τ ). N (Det(τ )) 2 g 2 ν∈S g Proof. This can be derived using the Poisson summation formula and we omit the detailed proof.
8.2. Joint weight enumerators and Hilbert–Siegel modular forms of genus g over K For given lattices Λ1 , . . . , Λg , let us consider the following theta-series ΘΛ1 ,...,Λg : H2g → C defined as: ΘΛ1 ,...,Λg (τ, z) =
e2πi Tr(σ ( xτ x 4 )) . (15) x∈Λ1 ×Λ2 ×···×Λg Here σ denotes the trace of the matrix, i.e., σ ((x)i, j ) = x ii . The next theorem states a connection between the theta-series defined over the lattices induced from codes over F4 and their joint weight enumerators. K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 649 Theorem 8.2. Let C j , 1 ≤ j ≤ g, be the codes over F4 and Λ j be an induced lattice from √ the codes C j , i.e., Λ j = (1/ 2)h̃ −1 (C j ). Let JC1 ,C2 ,...,C g (X) be the complete joint weight enumerator of the codes C j , 1 ≤ j ≤ g. Then the following holds: (g) ΘΛ1 ,Λ2 ,...,Λg (τ ) = JC1 ,C2 ,...,C g (θ1,µ (τ ) | µ ∈ S g ). Proof. Let h̃ : OnK × · · · × OnK → Fn4 × · · · × Fn4 be the homomorphism induced from the map h̃ in (10). For each v ∈ C1 × C2 × · · · × C g , let h̃ −1 (v) = h̃ −1 (0) + (ṽi ) be a preimage of v, all of whose entries (ṽi ) j = (ai j + bi j α) are of the forms such that 0 ≤ ai j , bi j < 2. Then
xτ x t e2πi Tr K /Q (σ ( 2 )) x∈ √1 h̃ −1 (v) 2 = e2πi Tr K /Q (σ ( (x 1 +ṽ1 ,...,x n +ṽn )τ (x 1 +ṽ1 ,...,x n +ṽn )t 4 )) x∈h̃ −1 (0)  = =

e x 1 ∈(2O K  (x +ṽ )τ (x +ṽ )t 2πi Tr K /Q ( 1 1 4 1 1 )g θ1,a(τ ) n a (ṽ1 ,...,ṽn ) ,  )  ···
x n ∈(2O K e t 2πi Tr K /Q ( (xn +ṽn )τ4(xn +ṽn ) )g  ) a∈S g g from the fact that the number of a in F4 which are equal to ṽ1 , . . . , ṽn is exactly n a (v1 , . . . , vn ).
Theorem 8.3. Let C j , 1 ≤ j ≤ g, be a length n Type II code over F4 . Let JC1 ,C2 ,...,C g (X) be the complete joint weight enumerator of the codes C j , 1 ≤ j ≤ g. Then (g) JC1 ,C2 ,...,C g (θ1,µ (τ ) | µ ∈ S g ) is a Hilbert–Siegel modular form of weight n/2 and genus g over K . (g) Proof. For simplicity, let H (τ ) := JC1 ,C2 ,...,C g (θ1,µ (τ ) | µ ∈ S g ). It is enough to check the transformation law of H (τ ) under the three types of generators of Γg (O K ) (see Remark 7.1):       1 α u 0 0 −I , ∀α ∈ Sym(g; O K ). ), , ∀u ∈ G L(g; O , K 0 1 0 u −1 I 0 Then, (g) H (−τ −1) = JC1 ,...,C g (θ2,µ (−τ −1 ) | µ ∈ S g ) g    0 −I χ   1 I 0 (g) N (Det(τ )) 2 (⊗g M)(θ1,µ (τ ) | µ ∈ S g ) = JC1 ,...,C g    g 2 650 K. Betsumiya, Y. Choie / European Journal of Combinatorics 26 (2005) 629–650 1 (g) JC ,...,C g ((⊗g M)(θ1,µ (τ, ) | µ ∈ S g )) 2gn 1 n 1 = N (Det(τ )) 2 g JC1 ,...,C g (τ ) i |Ci | n = N (Det(τ )) 2 n = N (Det(τ )) 2 H (τ ) (since n ≡ 0 (mod 4) and from MacWilliam’s identity given in Theorem 3.4).
9. Conclusion The interaction between coding theory, the theory of lattices and modular forms has been a source of many interesting results. Beginning with codes over fields and real lattices, it progressed to codes over F4 . Next the relationship with lattices and modular forms was developed (see [1] for example). It is well-known that the invariant ring, where the complete weight enumerators of the binary codes belong to, is isomorphic to the ring of elliptic modular forms by Broué–Enguehard. The present paper generalizes this relationship constructing the Hilbert modular form over K >. We develop the necessary coding theory over this field including the MacWilliam’s relations and investigate Type II codes. We use a set of weight enumerators over these √ codes to construct Hilbert Jacobi forms and Hilbert–Siegel modular forms over K = Q( 5). Acknowledgements The second author was partially supported by KRF 2003-070-C00001. References [1] E. Bannai, S.T. Dougherty, M. Harada, M. Oura, Type II codes, even unimodular lattices, and invariant rings, IEEE-IT 45 (4) (1999) 1194–1205. [2] Y. Choie, N. Kim, The complete weight enumerator of Type II code over Z4 and Jacobi forms, IEEE-IT 4 (1) (2001) 396–399. [3] Y. Choie, E. Lee, Jacobi forms over the totally real number fields and Codes over F p , Illinois J. Math. 46 (2) (2002) 627–643. [4] Y. Choie, P. Sole, Self-dual codes over Z 4 and half-integral weight modular forms, Proc. Amer. Math. Soc. 130 (11) (2002) 3125–3313. [5] W. Ebeling, Lattices and Codes, A course partially based on Lectures by F. Hirtzebruch, Advanced Lectures in Mathematics, Vieweg, 1994. [6] P. Gaborit, V. Pless, P. Solé, O. Atkin, Type II codes over F4 , Finite Fields Appl. 8 (2002) 171–183. [7] B. Liehl, On the group Sℓ2 over orders or arithmetic type, J. Reine Angew. Math. 323 (1981) 153–171. [8] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. [9] F.J. MacWilliams, A.M. Odlyzko, N.J.A. Sloane, Self-dual codes over G F(4), J. Combin. Theory Ser. A 25 (1978) 288–318. [10] H. Skogman, Jacobi forms over totally real number fields, Results Math. 39 (1–2) (2001) 169–182.