Self-dual codes over rings and the Chinese Remainder Theorem

Self-Dual Codes over Rings and the Chinese Remainder Theorem Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA Email: doughertys1@tiger.uofs.edu Masaaki Harada Department of Mathematical Sciences Yamagata University Yamagata 990–8560, Japan Email: harada@kszaoh3.kj.yamagata-u.ac.jp and Patrick Solé CNRS, I3S ESSI, BP 145 Route des Colles 06 903 Sophia Antipolis France Email: sole@essi.fr June 22, 2011 1 Abstract We give some characterizations of self-dual codes over rings, specifically the ring Z2k , where Z2k denotes the ring Z/2kZ of integers modulo 2k, using the Chinese Remainder Theorem, investigating Type I and Type II codes. The Chinese Remainder Theorem plays an important role in the study of self-dual codes over Z2k when 2k is not a prime power, while the Hensel lift is a powerful tool when 2k is a prime power. In particular, we concentrate on the case k = 3 and use construction A to build unimodular and 3-modular lattices. Keywords: Self-dual codes, codes over rings, unimodular lattices. 1991 Mathematical Subject Classification: Primary: 94B25, Secondary: 11H31 2 1 Introduction Self-dual codes over finite fields, especially binary and ternary fields, are a well studied subject, including their relationship to lattices and designs. Recently, codes over rings have increased in importance, generating much interest in these codes, for example see [1], [2], [3], [7], [8], [11], [15], [16] and [23]. In this paper, we give some characterizations of self-dual codes over rings, specifically the ring Z2k , where Z2k denotes the ring Z/2kZ of integers modulo 2k, using the Chinese Remainder Theorem. Recently, in [1] the notion of Type II codes over Z2k has been introduced. Here, we investigate Type II codes over Z2k using this theorem, giving special attention to the ring Z6 . We begin with some definitions. A code C over a ring R of length n is a subset of Rn , if it is an additive subgroup of Rn then it is called a linear code. In this paper all codes are assumed to be linear unless otherwise specified. An element of C is called a codeword of C. A generator matrix of C is a matrix whose rows generate C. We equip Rn with the P standard inner-product, i.e. [v, w] = vi wi . The orthogonal to a code is defined in the usual way, i.e. C ⊥ = {v ∈ Rn | [v, w] = 0 for all w ∈ C} where v = (v1 , v2 , . . . , vn ) and w = (w1 , w2 , . . . , wn ). We say that a code C is self-orthogonal if C ⊆ C ⊥ and C is self-dual if C = C ⊥ . MacWilliams relations for codes over any finite Frobenius ring are given in [23]. The paper is organized as follows. Section 2 gives some characterizations of self-dual codes over rings, specifically the ring Z2k . In Section 3, we pay attention to the ring Z6 . Some families of self-dual codes over Z6 (called senary codes) are also introduced. Section 4 deals with unimodular lattices corresponding to senary codes. In Sections 5 and 6, we investigate self-dual codes constructed from projective planes and weighing matrices. In the final section, we introduce new weight enumerators and establish their MacWilliams relations. 2 The Chinese Remainder Theorem and Self-Dual Codes Let R be a commutative ring (not necessarily finite) with a multiplicative identity denoted by 1. Let I1 , I2 , . . . , Ik be ideals of R such that: 1. Si = R/Ii is finite, 2. Ij + ∩k6=j Ik = R for 1 ≤ j ≤ k. That is, the ideals are relative prime, since R is commutative. Set I = ∩Ii and S = R/I. Define the map Ψ : S → (R/I1 ) × (R/I2 ) × · · · × (R/Ik ), by Ψ(α) = (α (mod I1 ), α (mod I2 ), . . . , α 3 (mod Ik )). The map Ψ−1 is a ring isomorphism by the generalized Chinese Remainder Theorem. Let C1 , C2 , . . . , Ck be codes where Ci is a code over Si , and define the code CRT (C1 , C2 , . . . , Ck ) = {Ψ−1 (v1 , v2 , . . . , vk ) | vi ∈ Ci }. We say that the code CRT (C1 , C2 , . . . , Ck ) is the Chinese product of codes C1 , C2 , . . . , Ck . Q It is clear that |CRT (C1 , C2 , . . . , Ck )| = ki=1 |Ci | and that if Ci is self-orthogonal for all i then CRT (C1 , C2 , . . . , Ck ) is self-orthogonal. This gives the following: Theorem 2.1 CRT (C1 , C2 , . . . , Ck ) is a self-dual code over S if and only if it is the Chinese product of self-dual codes C1 , . . . , Ck over S1 , . . . , Sk , respectively. We have the following restriction on the length of certain self-dual codes over Zk . e Corollary 2.2 Let s = pe11 · · · pj j where pi is prime for all i = 1, . . . , j. Suppose that there is at least one i such that ei =1. Then, if a self-dual code C of length n over Zs exists, n is even. In addition, if pi ≡ 3 (mod 4), then, if a self-dual code C of length n over Zs exists, n is a multiple of four. Proof. If there is a self-dual code of length n over the finite field Fp where p is prime then n is even. Since C is the Chinese product of a self-dual codes over Zpe11 , . . . , Zpej where at j least one Zpi is the finite field, the length n of C must be even. Moreover it is known that if there is a self-dual code of length n over Fp where p ≡ 3 (mod 4) then n is a multiple of four (cf. [20] and [21]). ✷ 2.1 Type II Codes over Z2k We begin by giving some characterizations of Type II codes over Z2k by the Chinese product. Recently, codes over Z4 have grown in importance. Interesting connections with binary codes and unimodular lattices have been found. Further connections have been found with codes over Z2k (cf. [1]). The connection between codes over Z4 and unimodular lattices prompted the definition of the Euclidean weight of a vector of Zn4 (cf. [2] and [3]). We defined the Euclidean weights of the elements 0, ±1, ±2, ±3, . . . , ±(k − 1), k of Z2k as 0, 1, 4, 9, . . ., (k − 1)2 , k 2 , respectively (cf. [1]). The Euclidean weight of a vector is just the rational sum of the Euclidean weights of its components. The Hamming weight of a vector is the number of non-zero components in the vector. We defined a Type II code over Z2k as a self-dual code with all codewords having Euclidean weight a multiple of 4k, see [1] for a complete discussion of these codes. If a self-dual code is not Type II, then it is said to be Type I. The notion of extremality for the Euclidean weight was also given in [1]. 4 Theorem 2.3 Let 2k = 2m r where r is odd. A code C is a Type II code over Z2k if and only if it is the Chinese product of a Type II code over Z2m and a self-dual code over Zr . Proof. If α ∈ Z2m r then there is a unique 0 ≤ β < 2m such that α = q2m + β for some integer q. This implies α ≡ β (mod 2m ) and, taking squares α2 ≡ β 2 (mod 2m+1 ), i.e. α2 ≡ (α (mod 2m ))2 (mod 2m+1 ). Then if v = (vi ) is a vector over Z2m r with Euclidean weight divisible by 2m+1 r, we have X if and only if both X and (vi X hold. vi2 ≡ 0 (mod 2m+1 r), (mod 2m ))2 ≡ 0 (vi (mod r))2 ≡ 0 (mod 2m+1 ), (mod r), ✷ The following corollary was shown in [1]. Here we give an alternative proof. Corollary 2.4 If there is a Type II code C of length n over Z2m r where r is odd, then n is a multiple of eight. e Proof. Let r = pe11 · · · pj j where pi is prime. Then C is the Chinese product of a self-dual code over Z2m and codes over rings Zpe11 , . . . , Zpei i . It is known in [8] that if there is a Type II code of length n over Z2m then n must be a multiple of eight. ✷ Recently the notion of shadow codes over Z4 has been introduced by the authors [10]. Here we consider shadow codes over Z2k . Similarly to Z4 , we pay attention to a certain subcode of index 2. The even weight subcode C0 of a Type I code C over Z2k is the set of codewords of C of Euclidean weights divisible by 4k. Lemma 2.5 The subcode C0 is Z2k −linear of index 2 in C. Proof. The first assertion follows by the self-duality of C using the relation (1) wE (x + y) = wE (x) + wE (y) + 2(x, y), 5 where wE (x) denotes the Euclidean weight of a vector x. The second assertion follows by observing that every codeword y of C has an Euclidean weight divisible by 2k. By the preceding relation we see that C2 := C − C0 is of the form x + C0 where x is any codeword of C of Euclidean weight congruent to 2k mod 4k and that translation by x is a one to one map from C0 onto C2 . ✷ S S By the preceding lemma we see that C is of index 2 in C0⊥ and we let C0⊥ = C C1 C3 . S With these notations define the shadow of C as S := C1 C3 . Unlike the binary case, C0⊥ /C0 is not necessarily isomorphic to the Klein 4-group, it may be isomorphic to either the Klein 4-group or the cyclic group of order 4. We now give some characterizations of shadow codes using the Chinese Remainder Theorem. Lemma 2.6 If k is an odd prime and C = CRT (B, K) with B a binary code and K a code over Zk then C0 = CRT (B0 , K), with B0 the even weight subcode of B, that is, the doubly-even subcode. Proof. Follows from the fact that the Euclidean weight of a vector x is divisible by 4k if and only if the Hamming weight of the binary vector (x (mod 2)) is doubly-even and the Euclidean weight of the vector x (mod k) over Zk is divisible by k, where x is an element of Z2k . ✷ Proposition 2.7 Let Sb be the shadow of B defined as B0⊥ = B and S = CRT (Sb , K). S Sb . Then C0⊥ = CRT (B0⊥ , K) Proof. Let x and y be elements of Zn2k , then it is easy to see that [x, y] = 0 if and only if [x (mod 2), y (mod 2)] = 0 and [x (mod k), y (mod k)] = 0. By Lemma 2.6, C0 = CRT (B0 , K). Thus C0⊥ = CRT (B0⊥ , K). Moreover S = C0⊥ − C is the same as CRT (B0⊥ − B, K) = CRT (Sb , K). ✷ 2.2 Codes over Polynomial Rings Let F be a finite field and let F[x] be the ring of polynomials over F. Let q(x) be a polynomial in F [x] such that the factorization of q(x) is given by: q(x) = p1 (x)p2 (x) · · · pr (x), where pi (x) is a non-constant irreducible polynomial and gcd(pi (x), pj (x)) ∈ F. Let Si = F [x]/(pi (x)), i.e. Si is a finite field, and S = F [x]/(q(x)), which is a field only if r = 1. Given codes over Si the Chinese product can be used to construct codes over S. This gives the following characterization of codes over a certain residue class ring F [x]/(q(x)). 6 Proposition 2.8 Let q(x) be a polynomial in F [x] such that the factorization of q(x) is given by: q(x) = p1 (x)p2 (x) · · · pr (x), where pi (x) is a non-constant irreducible polynomial and gcd(pi (x), pj (x)) ∈ F for i 6= j. Then a code over the ring F [x]/(q(x)) is the Chinese product of codes over some finite fields. Example 1 Let F = Z2 and let q(x) = x(x2 + x + 1). This gives that S1 is isomorphic to the finite field F2 with 2 elements and S2 is isomorphic to the finite field F4 , and S is a ring with 8 elements. 3 Senary Self-Dual Codes In this section we concentrate on self-dual codes over Z6 . The symmetrized weight enumerator (swe) of a senary code C is defined as: sweC (a, b, c, d) := X an0 (x) bn1 (x) cn2 (x) dn3 (x) , x∈C where ni (x) denotes the number of j such that xj = ±i. We say that two codes over Zk are equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. Codes differing by only a permutation of coordinates are called permutation-equivalent. 3.1 Some Families of Senary Self-Dual Codes We introduce a few families of senary self-dual codes together with Type II codes. Note that we can regard the lifted symmetry codes and the MacKay codes as bordered double circulant codes since the matrix W is a circulant matrix. 3.1.1 Extended Cyclic Codes The Chinese Remainder Theorem gives much information on the Chinese product codes. For example, if a permutation σ ∈ Sn is an automorphism of the Chinese product CRT (B, T ) then σ is also an automorphism of the binary code B and the ternary code T , where Sn is the symmetry group of degree n. In fact, the Chinese product of two cyclic codes is again a cyclic code. An extremal Type II code of length 24 was found in [1]. This code is an extended cyclic code of length 24 and the Chinese product of two extended cyclic quadratic residue codes over Z2 and Z3 . The next length for Type II codes is 32. There is no ternary self-dual code of length 32 with automorphism of order 31 (cf. [14]). Thus, this gives that there is no senary extended cyclic self-dual code of length 32. 7 3.1.2 Lifted Symmetry Codes We introduce families of double circulant codes. Of course, senary double circulant codes are constructed from binary and ternary double circulant codes. Here we describe a family of codes above the Pless symmetry codes. Let q be a prime power ≡ −1 (mod 6), and denote by χ the quadratic character of Fq . We begin by recalling some basic facts about the Jacobsthal matrix which hold more generally for any odd q. This matrix W = (Wi,j ) is indexed by the elements of Fq and has for a typical entry Wi,j := χ(j − i). The matrix W is instrumental in building Hadamard matrices of Paley type [17, Chap. II]. We collect here the properties that we need: (J1) JW = W J = 0 (J2) W W T = qI − J (J3) A := (J4) B := P i=✷ P i=✷ W−i,1 = −1 Wi,1 = 0 where J stands for the all-one matrix. See [17, Chap. II, Lemma 7] for proofs of (J1) and (J2). To prove (J3), (J4) observe firstly that by (J1) we have, knowing that −1 is not a quadratic residue, that A + B = −1. Secondly we have B= 1 2 X x∈Fq , x6=0 χ(1 − x2 ), and by the character property of χ B= 1 2 X x∈Fq , x6=0 χ(1 − x)χ(1 + x) = 0, the last equality coming from (J2). Now we define the matrix Sq as Sq =         0 1 ··· 1   χ(−1)  , ..  . W  χ(−1) which is q + 1 by q + 1 and satisfies Sq SqT = qI. Define a generator matrix of size q + 1 by 2q + 2 over Z6 by the rule G = ( I , Sq ). 8 Theorem 3.1 The matrix G generates a self-dual code P (n) of length n = 2q + 2 over Z6 . If furthermore q ≡ −1 (mod 12) then P (n) is Type II. Proof. By (J1) the rows of G are pairwise orthogonal. They are isotropic by the choice of q, since the inner-product of every row with itself is q + 1. Now in case q satisfies the congruence mod 12, the Euclidean weight of each row of G is divisible by 12 by the choice of q. This carries over to the row span by [1]. ✷ P (n) is the Chinese product of the ternary Pless symmetry code with generator matrix G and the binary self-dual codes with generator matrix ( I , J − I ). Thus we say that the above codes P (n) are the lifted symmetry codes. Of special interest are q = 5 yielding a Type I code above the Golay code, q = 17 yielding a Type I code of length 36, q = 11, 23 yielding Type II codes of lengths 24 and 48. We have obtained by computer that the symmetrized weight enumerators of the lifted symmetry codes P (12) and P (24) of lengths 12 and 24: sweP (12) = d12 + 24c6 d6 + 24c12 + 120bc8 d3 + 120b2 c4 d6 + 1280b3 c6 d3 +360b4 c8 + 1680b5 c4 d3 + 264b6 d6 + 768b6 c6 + 360b8 c4 + 440b9 d3 +24b12 + 240abc5 d5 + 1440ab2 c7 d2 + 960ab3 c3 d5 + 5760ab4 c5 d2 + 3360ab6 c3 d2 +120a2 c6 d4 + 2280a2 b2 c4 d4 + 3360a2 b3 c6 d + 1800a2 b4 c2 d4 + 5760a2 b5 c4 d + 1440a2 b7 c2 d +440a3 c9 + 960a3 bc5 d3 + 3360a3 b3 c3 d3 + 1680a3 b4 c5 + 960a3 b5 cd3 + 1280a3 b6 c3 +120a3 b8 c + 15a4 d8 + 1800a4 b2 c4 d2 + 2280a4 b4 c2 d2 + 120a4 b6 d2 + 960a5 b3 c3 d +240a5 b5 cd + 32a6 d6 + 264a6 c6 + 120a6 b4 c2 + 24a6 b6 + 15a8 d4 +a12 , sweP (24) = d24 + 1104c12 d12 + 48c24 + 3168bc20 d3 +23760b2 c16 d6 + 116160b3 c12 d9 + 23760b4 c8 d12 + 3168b4 c20 +3168b5 c4 d15 + 332640b5 c16 d3 + 4040256b6 c12 d6 + 1045440b7 c8 d9 +190080b8 c4 d12 + 23760b8 c16 + 4048b9 d15 + 7846080b9 c12 d3 +5844960b10 c8 d6 + 2059200b11 c4 d9 + 61824b12 d12 + 142656b12 c12 +3326400b13 c8 d3 + 3769920b14 c4 d6 + 242880b15 d9 + 23760b16 c8 +902880b17 c4 d3 + 198352b18 d6 + 3168b20 c4 + 24288b21 d3 +48b24 + 95040ab2 c19 d2 + 633600ab3 c15 d5 + 3326400ab4 c11 d8 +380160ab5 c7 d11 + 31680ab6 c3 d14 + 2661120ab6 c15 d2 + 41665536ab7 c11 d5 +9504000ab8 c7 d8 + 1013760ab9 c3 d11 + 28245888ab10 c11 d2 + 25470720ab11 c7 d5 +6177600ab12 c3 d8 + 5702400ab14 c7 d2 + 6031872ab15 c3 d5 + 601920ab18 c3 d2 +18480a2 c18 d4 + 95040a2 bc14 d7 + 342144a2 b2 c10 d10 + 31680a2 b3 c6 d13 +601920a2 b3 c18 d + 6177600a2 b4 c14 d4 + 28702080a2 b5 c10 d7 + 2566080a2 b6 c6 d10 +95040a2 b7 c2 d13 + 5702400a2 b7 c14 d + 142987680a2 b8 c10 d4 + 29367360a2 b9 c6 d7 9 +1672704a2 b10 c2 d10 + 28245888a2 b11 c10 d + 37224000a2 b12 c6 d4 + 5702400a2 b13 c2 d7 +2661120a2 b15 c6 d + 2827440a2 b16 c2 d4 + 95040a2 b19 c2 d + 3520a3 c9 d12 +24288a3 c21 + 443520a3 bc17 d3 + 1425600a3 b2 c13 d6 + 3125760a3 b3 c9 d9 +142560a3 b4 c5 d12 + 902880a3 b4 c17 + 22809600a3 b5 c13 d3 + 112464000a3 b6 c9 d6 +7223040a3 b7 c5 d9 + 102960a3 b8 cd12 + 3326400a3 b8 c13 + 212115200a3 b9 c9 d3 +41342400a3 b10 c5 d6 + 1013760a3 b11 cd9 + 7846080a3 b12 c9 + 22809600a3 b13 c5 d3 +1900800a3 b14 cd6 + 332640a3 b16 c5 + 443520a3 b17 cd3 + 3168a3 b20 c +66a4 d20 + 23760a4 c12 d8 + 47520a4 bc8 d11 + 2827440a4 b2 c16 d2 +9408960a4 b3 c12 d5 + 17083440a4 b4 c8 d8 + 475200a4 b5 c4 d11 + 37224000a4 b6 c12 d2 +216311040a4 b7 c8 d5 + 10216800a4 b8 c4 d8 + 34320a4 b9 d11 + 142987680a4 b10 c8 d2 +28036800a4 b11 c4 d5 + 190080a4 b12 d8 + 6177600a4 b14 c4 d2 + 190080a4 b15 d5 +18480a4 b18 d2 + 190080a5 c15 d4 + 380160a5 bc11 d7 + 494208a5 b2 c7 d10 +6031872a5 b3 c15 d + 28036800a5 b4 c11 d4 + 41817600a5 b5 c7 d7 + 696960a5 b6 c3 d10 +25470720a5 b7 c11 d + 216311040a5 b8 c7 d4 + 7223040a5 b9 c3 d7 + 41665536a5 b11 c7 d +9408960a5 b12 c3 d4 + 633600a5 b15 c3 d + 198352a6 c18 + 1900800a6 bc14 d3 +2566080a6 b2 c10 d6 + 1858560a6 b3 c6 d9 + 3769920a6 b4 c14 + 41342400a6 b5 c10 d3 +59000832a6 b6 c6 d6 + 475200a6 b7 c2 d9 + 5844960a6 b8 c10 + 112464000a6 b9 c6 d3 +2566080a6 b10 c2 d6 + 4040256a6 b12 c6 + 1425600a6 b13 c2 d3 + 23760a6 b16 c2 +15840a7 c9 d8 + 5702400a7 b2 c13 d2 + 7223040a7 b3 c9 d5 + 3611520a7 b4 c5 d8 +29367360a7 b6 c9 d2 + 41817600a7 b7 c5 d5 + 166320a7 b8 cd8 + 28702080a7 b10 c5 d2 +380160a7 b11 cd5 + 95040a7 b14 cd2 + 495a8 d16 + 190080a8 c12 d4 +166320a8 bc8 d7 + 6177600a8 b3 c12 d + 10216800a8 b4 c8 d4 + 3611520a8 b5 c4 d7 +9504000a8 b7 c8 d + 17083440a8 b8 c4 d4 + 15840a8 b9 d7 + 3326400a8 b11 c4 d +23760a8 b12 d4 + 242880a9 c15 + 1013760a9 bc11 d3 + 475200a9 b2 c7 d6 +2059200a9 b4 c11 + 7223040a9 b5 c7 d3 + 1858560a9 b6 c3 d6 + 1045440a9 b8 c7 +3125760a9 b9 c3 d3 + 116160a9 b12 c3 + 1672704a10 b2 c10 d2 + 696960a10 b3 c6 d5 +2566080a10 b6 c6 d2 + 494208a10 b7 c2 d5 + 342144a10 b10 c2 d2 + 34320a11 c9 d4 +1013760a11 b3 c9 d + 475200a11 b4 c5 d4 + 380160a11 b7 c5 d + 47520a11 b8 cd4 +2972a12 d12 + 61824a12 c12 + 102960a12 bc8 d3 + 190080a12 b4 c8 +142560a12 b5 c4 d3 + 23760a12 b8 c4 + 3520a12 b9 d3 + 1104a12 b12 +95040a13 b2 c7 d2 + 31680a13 b6 c3 d2 + 31680a14 b3 c6 d + 4048a15 c9 +3168a15 b4 c5 + 495a16 d8 + 66a20 d4 + a24 . 10 3.1.3 MacKay Codes Let q be a prime power congruent to −5 (mod 12). Define a matrix Bq bordering Jacobsthal as   2 1 ··· 1     1   . Bq =  .  .. W − 2I    1 A double circulant code Mq is then introduced by its generator matrix ( I , Bq ). Theorem 3.2 The MacKay code Mq is a self-dual Type II code of length 2q + 2. Proof. Follows from the property of W which is skew-symmetric for q ≡ −1 Observe that the inner-product of each row with itself is this time q + 5. (mod 4). ✷ Remark. Mq is the Chinese product of the binary doubly-even self-dual code with generator matrix ( I , J − I ) and the ternary self-dual code with generator matrix ( I , Bq ). In particular q = 19 yields after the real construction A6 an extremal lattice in dimension 40 [6]. For the first case q = 7, we have found its symmetrized weight enumerator: sweM7 = d16 + 480bc8 d7 + 1792b3 c12 d + 24864b4 c8 d4 + 5376b5 c4 d7 +224b6 d10 + 25344b7 c8 d + 25536b8 c4 d4 + 2720b9 d7 + 5376b11 c4 d +3360b12 d4 + 256b15 d + 256ac15 + 2688abc11 d3 + 10752ab2 c7 d6 +896ab3 c3 d9 + 5376ab4 c11 + 166656ab5 c7 d3 + 24192ab6 c3 d6 + 25344ab8 c7 +45696ab9 c3 d3 + 1792ab12 c3 + 20160a2 b2 c10 d2 + 72576a2 b3 c6 d5 + 2016a2 b4 c2 d8 +282240a2 b6 c6 d2 + 32256a2 b7 c2 d5 + 20160a2 b10 c2 d2 + 1568a3 c9 d4 + 2688a3 bc5 d7 +45696a3 b3 c9 d + 185472a3 b4 c5 d4 + 2688a3 b5 cd7 + 166656a3 b7 c5 d + 12768a3 b8 cd4 +2688a3 b11 cd + 28a4 d12 + 3360a4 c12 + 12768a4 bc8 d3 + 11424a4 b2 c4 d6 +25536a4 b4 c8 + 185472a4 b5 c4 d3 + 672a4 b6 d6 + 24864a4 b8 c4 + 1568a4 b9 d3 +32256a5 b2 c7 d2 + 16128a5 b3 c3 d5 + 72576a5 b6 c3 d2 + 672a6 c6 d4 + 24192a6 b3 c6 d +11424a6 b4 c2 d4 + 10752a6 b7 c2 d + 2720a7 c9 + 2688a7 bc5 d3 + 5376a7 b4 c5 +2688a7 b5 cd3 + 480a7 b8 c + 198a8 d8 + 2016a8 b2 c4 d2 + 896a9 b3 c3 d +224a10 c6 + 28a12 d4 + a16 . 11 3.1.4 A Family of Type II Codes It is well known that there is a unique binary doubly-even self-dual code B and there is a unique ternary self-dual code T of length 8, up to equivalence. B and T have the following generator matrices: GB =        1000 0100 0010 0001 0111 1011 1101 1110        and GT =        1000 0100 0010 0001 1200 1100 0012 0011     ,   respectively. A generator matrix of the Chinese product CRT (B, T ) of B and T is        1000 0100 0010 0001  4533 1433 3345 3314    .   The symmetrized weight enumerator of the senary code CRT (B, T ) is sweCRT (B,T ) = d8 + 192b2 c4 d2 + 16b3 d5 + 64b6 d2 + 16ac3 d4 + 512ab3 c3 d + 64a2 c6 +96a2 bc2 d3 + 192a2 b4 c2 + 96a3 b2 cd2 + 14a4 d4 + 16a4 b3 d + 16a5 c3 + a8 . ⊕n CRT (B, T ) is a Type II code of length 8n whose symmetrized weight enumerator is sweCRT (B,T ) n . 3.2 Properties of Senary Self-Dual Codes Any code over Z6 is permutation-equivalent to a code generated by the following matrix: (2)   Ik A1,2 A1,3 A1,4  1   0 2I  k2 2A2,3 2A2,4  ,  0 0 3Ik3 3A3,4 where Ai,j are binary matrices for i > 1. Such a code is said to have rank {1k1 , 2k2 , 3k3 }, see [1]. Lemma 3.3 Let C be a senary code of rank {1k1 , 2k2 , 3k3 }. If C is a self-dual code of length n then k2 = k3 and k1 + k2 = n/2. n Proof. A senary self-dual code of length n has 6 2 codewords. A code of rank {k1 , k2 , k3 } has 6k1 3k2 2k3 codewords. Hence, if the code is self-dual then k2 must be equal to k3 otherwise the number of codewords would not be a multiple of 6. Moreover k1 + k2 = n2 , since 12 6k1 3k2 2k2 = 6k1 (3 · 2)k2 . ✷ If C is a code over Z6 , let C2 be the code read (mod 2) and let C3 be the code read (mod 3). That is, C2 = {v|v ≡ w (mod 2), w ∈ C}, and C3 = {v|v ≡ w (mod 3), w ∈ C}. The code C2 is permutation-equivalent to a code with generator matrix of the form: (3)   I A1,2 A1,3 A1,4   k1 , 0 0 3Ik3 3A3,4 where Ai,j are binary matrices for i > 1. Notice 3 ≡ 1 (mod 2) hence this code generates a binary code of dimension k1 + k3 = n2 . And the ternary code C3 is permutation-equivalent to a code with generator matrix of the form: (4)   I A1,2 A1,3 A1,4   k1 , 0 2Ik2 2A2,3 2A2,4 where Ai,j are binary matrices for i > 1. Notice 2 is a unit in Z3 hence this code generates a ternary code of dimension k1 + k2 = n2 . We now consider self-dual codes of length n over Z6 constructed from a fixed binary self-dual code C2 and a fixed ternary self-dual code C3 by the Chinese product. Let T be the set of all codes constructed by permuting the coordinates of C2 and C3 . Let C3′ be a ternary code obtained from C3 by changing the signs of certain coordinates, then it is clear that CRT (C2 , C3 ) is equivalent to CRT (C2 , C3′ ). Moreover CRT (C2 α , C3 β ) is equivalent to −1 CRT (C2 α(β) , C3 ) where α and β are element of the symmetric group Sn of degree n. Of course, Sn acts on the coordinates of C2 and C3 . Hence T = {CRT (C2 α , C3 β ) | α, β ∈ Sn } = {CRT (C2 γ , C3 ) | γ ∈ Sn }. In addition, if γ is an element of the automorphism group Aut(C2 ) of C2 then CRT (C2 , C3 ) = CRT (C2 γ , C3 ). Therefore the number N of inequivalent codes obtained from C2 and C3 by permuting the coordinates and changing the signs is at most n! , |Aut(C2 )| where |Aut(C2 )| denotes the order of Aut(C2 ). 13 Let Aut(C3 ) be the group of all permutations which preserve C3 , similarly, we have N≤ Thus we have n! . |Aut(C3 )| ( ) n! n! N ≤ min . , |Aut(C2 )| |Aut(C3 )| This gives the following upper bound on the number of inequivalent senary self-dual codes. Proposition 3.4 Let C2 and C3 be the sets of all inequivalent self-dual codes of length n over Z2 and Z3 , respectively. Let N6 (n) be the number of inequivalent self-dual codes of length n over Z6 . Then N6 (n) is bounded by (5)    N6 (n) ≤ min |C3 |   X C2 ∈C2     X n! n!  .  , |C2 |   |Aut(C2 )| C3 ∈C3 |Aut(C3 )| We give a classification of self-dual codes over Z6 of length 4. By Lemma 3.3, the rank of a self-dual code of length 4 is either {12 } or {11 , 21 , 31 }. When rank is {12 }, any code is equivalent to a code with generator matrix of the form:   1 0 1 2   . 0 1 4 1 When rank is {11 , 21 , 31 }, it is easy to see that a generator matrix of a self-dual code can be transformed into a matrix of the form:   1 a b c    0 2 1 1 ,   0 0 3 3 where a, b, c ∈ Z6 . We found all self-dual codes by finding all possible (a, b, c). Then any code of rank {11 , 21 , 31 } is equivalent to a code with generator matrix of the form:   1 1 0 4    0 2 1 1 .   0 0 3 3 Therefore there are exactly two inequivalent self-dual codes of length 4. Since N6 (4) ≤ 3, the above bound (5) is always not tight in general. All binary self-dual codes of length up to 30 and all ternary self-dual codes of length up to 20 have been classified (cf. [4] and [22]). It would be interesting to determine equivalence classes of senary self-dual codes of length up to 20 from these codes. 14 4 Corresponding Lattices An n-dimensional lattice Λ in Rn is the set of integer linear combinations of n linearly independent vectors v1 , . . . , vn , where Rn is the n-dimensional Euclidean space. The dual lattice Λ∗ is given by Λ∗ = {x ∈ Rn |[x, a] ∈ Z for all a ∈ Λ}, where [x, a] = x1 a1 +· · ·+xn an and x = (x1 , . . . , xn ), a = (a1 , . . . , an ). A lattice Λ is integral if the inner product of any two lattice points is integral, or equivalently, if Λ ⊆ Λ∗ . An integral lattice with Λ = Λ∗ is called unimodular. The theta series θΛ (q) of a lattice Λ is the formal power series θΛ (q) = X q [x,x] . x∈Λ The kissing number is the first non-trivial coefficient of the theta series. 4.1 Construction A6 Every senary code C can be attached a lattice by the formula 1 A6 (C) = √ (C + 6Zn ). 6 Using that construction the Leech lattice was constructed anew in [1]. If C is self-dual then A6 (C) is unimodular, moreover if C is Type II then A6 (C) is even unimodular (cf. [1]). We observe that construction B3 of Leech and Sloane [6, p.148] is in fact construction A6 applied to the code S = −2C + 3Pn , where C is a ternary code and Pn is the binary parity-check code of length n. In other words, S is the Chinese product of C by Pn . Let νi denote the theta series of Z + 6i for i = 1, 2, 3. Clearly θA6 (C) = sweC (θ3 , ν1 , ν2 , ν3 ). With the denotations of [6, p.105] we have ν1 = ψ6 , ν2 = ψ3 , ν3 = θ2 . 4.2 Even Unimodular Lattices In Section 3, we gave a family of Type II codes ⊕n CRT (B, T ) of length 8n. Since there is a unique 8-dimensional even unimodular lattice, up to equivalence, namely E8 , A6 (CRT (B, T )) must be E8 . In addition, it is easy to see that A6 (⊕n CRT (B, T )) is E8 + · · · + E8 . The minimum norm of the lattice A6 (P (12)) is 1, the kissing number is 24 and the lattice is a unimodular lattice. Thus A6 (P (12)) is Z12 . Since the code M7 of length 16 is Type II and dE = 12, A6 (M7 ) is a 16-dimensional even extremal unimodular lattice, that is, either + E8 + E8 or D16 by Table 16.7 of [6]. The lattice A6 (P (24)) is a 24-dimensional even unimodular lattice. Moreover, from sweP (24) of P (24) in Section 3, the theta series θA6 (P (24)) of the lattice A6 (P (24)) is 1 + + 1104q 2 + · · ·. Thus the lattice is D24 by Table 16.1 in [6]. 15 4.3 Odd Unimodular Lattices From [5, Table II], there is a unique odd unimodular lattice with the minimum norm 2 in dimensions 12 and 16. By the Chinese product, we have Type I codes C12 and C16 of lengths 12 and 16 with the following generator matrices             100000 010000 001000 000100 000010 000001 231155 141113 112355 151235 513545 355112              and                  10000000 01000000 00100000 00010000 00001000 00000100 00000010 00000001 44411111 42521511 42512115 42151455 44511525 42115554 14422242 41222424          ,         respectively. Their symmetrized weight enumerators swe12 and swe16 are sweC12 = d12 + 24c12 + 120bc8 d3 + 120b2 c4 d6 + 1280b3 c6 d3 +360b4 c8 + 1680b5 c4 d3 + 264b6 d6 + 768b6 c6 + 360b8 c4 +440b9 d3 + 24b12 + 384abc5 d5 + 1440ab2 c7 d2 + 960ab3 c3 d5 +5760ab4 c5 d2 + 3360ab6 c3 d2 + 120a2 c6 d4 + 1920a2 b2 c4 d4 + 3360a2 b3 c6 d +1800a2 b4 c2 d4 + 5760a2 b5 c4 d + 1440a2 b7 c2 d + 440a3 c9 + 960a3 bc5 d3 +3840a3 b3 c3 d3 + 1680a3 b4 c5 + 960a3 b5 cd3 + 1280a3 b6 c3 + 120a3 b8 c +15a4 d8 + 1800a4 b2 c4 d2 + 1920a4 b4 c2 d2 + 120a4 b6 d2 + 960a5 b3 c3 d +384a5 b5 cd + 32a6 d6 + 264a6 c6 + 120a6 b4 c2 + 15a8 d4 +a12 , sweC16 = d16 + 40c12 d4 + 144bc8 d7 + 64bc14 d + 40b2 c4 d10 +1712b2 c10 d4 + 1760b3 c6 d7 + 1024b3 c12 d + 56b4 c2 d10 + 12304b4 c8 d4 +2992b5 c4 d7 + 5952b5 c10 d + 224b6 d10 + 24032b6 c6 d4 + 1632b7 c2 d7 +12032b7 c8 d + 14632b8 c4 d4 + 2720b9 d7 + 9920b9 c6 d + 3696b10 c2 d4 +3072b11 c4 d + 3360b12 d4 + 448b13 c2 d + 256b15 d + 136ac9 d6 +256ac15 + 112abc5 d9 + 1376abc11 d3 + 5024ab2 c7 d6 + 448ab2 c13 +352ab3 c3 d9 + 22912ab3 c9 d3 + 18640ab4 c5 d6 + 3072ab4 c11 + 224ab5 cd9 +77120ab5 c7 d3 + 14144ab6 c3 d6 + 9920ab6 c9 + 82432ab7 c5 d3 + 2856ab8 cd6 +12032ab8 c7 + 25952ab9 c3 d3 + 5952ab10 c5 + 2688ab11 cd3 + 1024ab12 c3 +64ab14 c + 2a2 d14 + 104a2 c6 d8 + 336a2 c12 d2 + 4056a2 bc8 d5 +1152a2 b2 c4 d8 + 11760a2 b2 c10 d2 + 35200a2 b3 c6 d5 + 1392a2 b4 c2 d8 + 77232a2 b4 c8 d2 +55536a2 b5 c4 d5 + 168a2 b6 d8 + 136992a2 b6 c6 d2 + 17952a2 b7 c2 d5 + 77232a2 b8 c4 d2 +952a2 b9 d5 + 11760a2 b10 c2 d2 + 336a2 b12 d2 + 816a3 c9 d4 + 1152a3 bc5 d7 +2688a3 bc11 d + 26336a3 b2 c7 d4 + 4192a3 b3 c3 d7 + 25952a3 b3 c9 d + 87472a3 b4 c5 d4 16 +1376a3 b5 cd7 + 82432a3 b5 c7 d + 62176a3 b6 c3 d4 + 77120a3 b7 c5 d + 7616a3 b8 cd4 +22912a3 b9 c3 d + 1376a3 b11 cd + 16a4 d12 + 360a4 c6 d6 + 3360a4 c12 +7616a4 bc8 d3 + 4984a4 b2 c4 d6 + 3696a4 b2 c10 + 62176a4 b3 c6 d3 + 5432a4 b4 c2 d6 +14632a4 b4 c8 + 87472a4 b5 c4 d3 + 424a4 b6 d6 + 24032a4 b6 c6 + 26336a4 b7 c2 d3 +12304a4 b8 c4 + 816a4 b9 d3 + 1712a4 b10 c2 + 40a4 b12 + 952a5 c9 d2 +2640a5 bc5 d5 + 17952a5 b2 c7 d2 + 8576a5 b3 c3 d5 + 55536a5 b4 c5 d2 + 2640a5 b5 cd5 +35200a5 b6 c3 d2 + 4056a5 b8 cd2 + 62a6 d10 + 424a6 c6 d4 + 2856a6 bc8 d +5432a6 b2 c4 d4 + 14144a6 b3 c6 d + 4984a6 b4 c2 d4 + 18640a6 b5 c4 d + 360a6 b6 d4 +5024a6 b7 c2 d + 136a6 b9 d + 2720a7 c9 + 1376a7 bc5 d3 + 1632a7 b2 c7 +4192a7 b3 c3 d3 + 2992a7 b4 c5 + 1152a7 b5 cd3 + 1760a7 b6 c3 + 144a7 b8 c +94a8 d8 + 168a8 c6 d2 + 1392a8 b2 c4 d2 + 1152a8 b4 c2 d2 + 104a8 b6 d2 +224a9 bc5 d + 352a9 b3 c3 d + 112a9 b5 cd + 62a10 d6 + 224a10 c6 +56a10 b2 c4 + 40a10 b4 c2 + 16a12 d4 + 2a14 d2 + a16 . Thus the minimum Euclidean weights of both C12 and C16 are 12. A6 (C12 ) and A6 (C16 ) are the odd unimodular lattices with the minimum norm 2. 4.4 A 3-modular Lattice Recently Gabriele Nebe [18] has found an extremal 3-modular lattice in dimension 24 from a code N over Z6 . N has the following generator matrix:                              021504155450211000001131 011045013540150005055212 402051511014011450502145 121041050541020015551212 520540110455511555555300 511150105450203514055511 333300000000000000000000 000033330000000000000000 000000003333000000000000 000000000000333300000000 000000000000000033330000 000000000000000000003333                .              We have obtained its symmetrized weight enumerator: sweN = d24 + 626b5 c16 d3 + 5668b6 c12 d6 + 3060b7 c8 d9 + 2b8 c4 d12 +6b9 d15 + 5982b9 c12 d3 + 12826b10 c8 d6 + 130b11 c4 d9 + 26b12 d12 +5626b13 c8 d3 + 280b14 c4 d6 + 178b15 d9 + 42b17 c4 d3 + 154b18 d6 +12b21 d3 + 6ab2 c19 d2 + 1428ab3 c15 d5 + 4594ab4 c11 d8 + 670ab5 c7 d11 17 +3926ab6 c15 d2 + 49150ab7 c11 d5 + 21474ab8 c7 d8 + 38ab9 c3 d11 + 23002ab10 c11 d2 +67556ab11 c7 d5 + 428ab12 c3 d8 + 10930ab14 c7 d2 + 410ab15 c3 d5 + 24ab18 c3 d2 +190a2 bc14 d7 + 472a2 b2 c10 d10 + 286a2 b3 c6 d13 + 24a2 b3 c18 d + 12546a2 b4 c14 d4 +29094a2 b5 c10 d7 + 5518a2 b6 c6 d10 + 18a2 b7 c2 d13 + 8114a2 b7 c14 d + 170184a2 b8 c10 d4 +73758a2 b9 c6 d7 + 74a2 b10 c2 d10 + 22362a2 b11 c10 d + 93322a2 b12 c6 d4 + 382a2 b13 c2 d7 +4310a2 b15 c6 d + 188a2 b16 c2 d4 + 6a2 b19 c2 d + 134a3 c9 d12 + 780a3 c21 +46a3 bc17 d3 + 2630a3 b2 c13 d6 + 2748a3 b3 c9 d9 + 698a3 b4 c5 d12 + 42a3 b4 c17 +54280a3 b5 c13 d3 + 124326a3 b6 c9 d6 + 14926a3 b7 c5 d9 + 18a3 b8 cd12 + 5754a3 b8 c13 +252276a3 b9 c9 d3 + 100798a3 b10 c5 d6 + 42a3 b11 cd9 + 5854a3 b12 c9 + 59400a3 b13 c5 d3 +102a3 b14 cd6 + 882a3 b16 c5 + 46a3 b17 cd3 + 6a4 d20 + 4a4 c12 d8 +56a4 bc8 d11 + 188a4 b2 c16 d2 + 21404a4 b3 c12 d5 + 17494a4 b4 c8 d8 + 2430a4 b5 c4 d11 +78730a4 b6 c12 d2 + 241014a4 b7 c8 d5 + 21738a4 b8 c4 d8 + 172104a4 b10 c8 d2 + 70556a4 b11 c4 d5 +15490a4 b14 c4 d2 + 26a4 b15 d5 + 26a5 c15 d4 + 1190a5 bc11 d7 + 842a5 b2 c7 d10 +410a5 b3 c15 d + 57884a5 b4 c11 d4 + 39924a5 b5 c7 d7 + 2390a5 b6 c3 d10 + 58340a5 b7 c11 d +247798a5 b8 c7 d4 + 16494a5 b9 c3 d7 + 46846a5 b11 c7 d + 23836a5 b12 c3 d4 + 1940a5 b15 c3 d +8730a6 c18 + 102a6 bc14 d3 + 4650a6 b2 c10 d6 + 3340a6 b3 c6 d9 + 280a6 b4 c14 +90174a6 b5 c10 d3 + 51084a6 b6 c6 d6 + 1956a6 b7 c2 d9 + 12570a6 b8 c10 + 127142a6 b9 c6 d3 +5034a6 b10 c2 d6 + 4260a6 b12 c6 + 2630a6 b13 c2 d3 + 20a7 c9 d8 + 382a7 b2 c13 d2 +11886a7 b3 c9 d5 + 5640a7 b4 c5 d8 + 62494a7 b6 c9 d2 + 42356a7 b7 c5 d5 + 598a7 b8 cd8 +31910a7 b10 c5 d2 + 1062a7 b11 cd5 + 318a7 b14 cd2 + 751a8 d16 + 598a8 bc8 d7 +428a8 b3 c12 d + 17002a8 b4 c8 d4 + 5128a8 b5 c4 d7 + 19042a8 b7 c8 d + 16726a8 b8 c4 d4 +20a8 b9 d7 + 4722a8 b11 c4 d + 4a8 b12 d4 + 10290a9 c15 + 42a9 bc11 d3 +1316a9 b2 c7 d6 + 130a9 b4 c11 + 13262a9 b5 c7 d3 + 3084a9 b6 c3 d6 + 2804a9 b8 c7 +2108a9 b9 c3 d3 + 74a10 b2 c10 d2 + 1622a10 b3 c6 d5 + 4366a10 b6 c6 d2 + 458a10 b7 c2 d5 +216a10 b10 c2 d2 + 38a11 b3 c9 d + 1406a11 b4 c5 d4 + 670a11 b7 c5 d + 56a11 b8 cd4 +2452a12 d12 + 2970a12 c12 + 18a12 bc8 d3 + 2a12 b4 c8 + 442a12 b5 c4 d3 +6a12 b9 d3 + 18a13 b2 c7 d2 + 30a13 b6 c3 d2 + 134a15 c9 + 879a16 d8 +6a20 d4 + a24 . 5 Self-Dual Codes over Rings formed from Projective Planes Let Π be a projective plane of order n = Case 1: Each pi = 2 or pi ≡ 1 (mod 4) Case 2: Each pi ≡ 3 (mod 4). Qr i=1 pi where the pi are distinct primes with either: or Let Cpi be the self-code over Fpi of length n2 + n + 2 or n2 + n + 4 depending on the case 18 formed as given in [9]. Q Theorem 5.1 Let Π be a projective plane of order n = pi where the pi are distinct primes with the above cases, then CRT (Cp1 , Cp2 , . . . , Cpr ) is a self-dual code over the ring Zp1 p2 ...pr of length n2 + n + 2 or n2 + n + 4 for Case 1 and Case 2 respectively. Proof. Since each Cpi is a self-dual code over Fpi of the same length then the Chinese product gives that the code CRT (Cp1 , Cp2 , . . . , Cpr ) is self-dual. ✷ Q Note for Case 2, r must be odd. If r = 2k then n ≡ pi ≡ 32k ≡ 1 (mod 4). Hence n2 + n + 1 ≡ 3 (mod 4) and then n2 + n + 4 ≡ 2 (mod 4) giving that there are no self-dual codes over Fpi . Corollary 5.2 If n = 2p where p is a prime and p ≡ 1 (mod 4) then a Type II self-dual code of length N = n2 + n + 3 ± 1 can be constructed from a projective plane of order n over Z2p . The N −dimensional even unimodular lattice obtained by Construction A has minimum norm 2. Of course, there are no known non-trivial examples of either Case 1 or Case 2 since all known planes known have orders a power of a prime. Had a projective plane of order 10 existed its attached lattice in dimension 112 = 8.14 would have had by [6, Chap.17, Theorem 7]. A theta series of the shape: θ10 (q) = E414 + 4 X ai E414−3i ∆i , i=1 where letting t = q 2 , we denote by E4 = 1 + 240 t + 2160 t2 + 6720 t3 + 17520 t4 + 30240 t5 + · · · , the theta series of the E8 lattice, and by ∆ the cusp form of weight 12 for the full modular group Y ∆ = t (1 − tr )24 , r≥1 or up to order 5 ∆ = t − 24 t2 + 252 t3 − 1472 t4 + 4830 t5 . 6 Weighing Matrices and Type II Codes In this section, we deal with weighing matrices corresponding to Type II codes. A weighing matrix W (m, k) of order m and weight k is an m by m (0,1,−1)-matrix such that W · W T = kIm , k ≤ m. A weighing matrix W (m, m) is just a Hadamard matrix. 19 Weighing matrices are a generalization of Hadamard matrices. We say that two weighing matrices W1 and W2 of order m and weight k are equivalent if there exist monomial matrices of 0’s, 1’s and −1’s P and Q such that W1 = P · W2 · Q. We give a method for constructing self-dual codes over Z2n . Theorem 6.1 Let x an element of Z2n satisfying 1 + x2 k ≡ 0 (mod 2n). Let Wm,k be a weighing matrix of order m and weight k. Then the matrix G = ( Im , xWm,k ) generates a self-dual code C over Z2n of length 2m. Moreover if x satisfies 1 + x2 k ≡ 0 (mod 4n) then C is Type II. Proof. G · GT = (1 + x2 k)Im . Thus if 1 + x2 k ≡ 0 if 1 + x2 k ≡ 0 (mod 4n) then C is Type II. (mod 2n) then C is self-dual. Moreover ✷ Remark. For n = 2, this method was given in [12]. Since the matrix Sq in the generator matrix of the lifted senary symmetry codes is a weighing matrix of order q + 1 and weight q, this method is a generalization of Theorem 3.1. Example 2 All weighing matrices have been classified for order 12 (cf. [19]). There are weighing matrices W12,k of order 12 for every weight 1 ≤ k ≤ 12. For n = 3, the matrix ( I , W12,11 ) generates a Type II code of length 24 with the minimum Euclidean weight 12, that is, this code is not extremal. Since there is a unique weighing matrix of weight 11, this is the same code given previously. Example 3 For n = 4, the matrix ( I , 3W12,7 ) generates a Type II code of length 24. There are exactly three inequivalent weighing matrices of weight 7. The three inequivalent matrices are denoted by A1 , A3 and A8 in [19]. Since the matrix A1 has the intersection pattern p6 ≥ 1 (for the definition see [19]), the Type II code with generator matrix of the form ( I , 3A1 ) contains a codeword of Euclidean weight 16, that is the code is not extremal. Moreover we have verified by computer that the codes constructed from the remaining two weighing matrices are not extremal. It was shown in [1] that a Type II code of length n exists if and only if n ≡ 0 Thus we have the following restriction on the existence of weighing matrices. (mod 8). Corollary 6.2 Suppose that there is a weighing matrix of order m and weight k. If there is an element x of Z2n satisfying 1 + x2 k ≡ 0 (mod 4n) for certain n, then m ≡ 0 (mod 4). Remark. For n = 1, the above corollary was shown in [13]. As a corollary to Theorem 6.1, we have the following: 20 Corollary 6.3 Let α and β be elements of Z2n satisfying 1 + α2 + β 2 k ≡ 0 (mod 2n). T If either Wm,k is a skew-symmetric weighing matrix (that is, Wm,k = −Wm,k ) or Wm,k is a symmetric weighing matrix with αβ ≡ 0 (mod n), then the matrix G = ( I , (αI +βWm,k ) ) generates a self-dual code C over Z2n of length 2m. Moreover if 1 + α2 + β 2 k ≡ 0 (mod 4n) then C is Type II. T Proof. We have G · GT = (1 + α2 + kβ 2 )I + αβWm,k + αβWm,k . It follows from the assumptions that G · GT = 0. ✷ Remark. We can regard MacKay codes over Z6 described in Section 3 as a special case of the above corollary. A similar argument to Corollary 6.2 gives the following: Corollary 6.4 Suppose that there are two elements α and β of Z2n satisfying 1+α2 +β 2 k ≡ 0 (mod 4n) for certain n where β 6= 0. (1) If there is a skew-symmetric weighing matrix of order m and weight k, then m ≡ 0 (mod 4). (2) If there is a symmetric weighing matrix of order m and weight k and αβ ≡ 0 then m ≡ 0 (mod 4). 7 (mod n), The Complete Combined Weight Enumerator In this section all rings will assumed to be commutative, finite and Frobenius. Let C1 , C2 , . . . , Cs be codes of length n, where Ci is a code over the ring Ri . Let C = C1 × C2 × · · · × Cs and R = R1 × R2 × · · · × Rs . Definition 1 The complete combined weight enumerator is given by: X P (C1 , C2 , . . . , Cs )(Xa ) = Y Xana (c1 ,c2 ,...,cs ) , (c1 ,c2 ,...,cs )∈C a∈R where na (c1 , c2 , . . . , cs ) = |{i|a = (ci1 , cis , . . . , cis )}| and cij is the i-th coordinate of cj . 7.1 The MacWilliams Relations We will prove the MacWilliams relations for this new weight enumerator by generalizing the technique in [1], which itself is a generalization of [23]. Let Gi be a group with f : G1 × G2 × · · · × Gs → A where A is a complex algebra. Denote c the character group of G , that is G c = {π|π is a character of G } where a character by G i i i i of a group is a group homomorphism from G to the Complex numbers under multiplication. 21 c ×G c × ··· × G c → A by Define fb : G 1 2 s fb(π1 , π2 , . . . , πs ) = X X x1 ∈G1 x2 ∈G2 ··· X xs ∈Gs π1 (x1 )π2 (x2 ) · · · πs (xs )f (x1 , x2 , . . . , xs ). Lemma 7.1 The function f (x1 , x2 , . . . , xs ) = X X X 1 ··· π1 (−x1 )π2 (−x2 ) · · · πs (−xs )fb(π1 , π2 , . . . , πs ). |G1 ||G2 | · · · |Gs | c c c π1 ∈ G 1 π1 ∈ G 2 π1 ∈ G s Proof. We have that X X X 1 ··· π1 (−x1 )π2 (−x2 ) · · · πs (−xs )fb(π1 , π2 , . . . , πs ) |G1 ||G2 | · · · |Gs | c1 π2 ∈G c2 cs π1 ∈ G πs ∈ G X X 1 π1 (−x1 )π2 (−x2 ) . . . πs (−xs ) ··· = |G1 ||G2 | · · · |Gs | cs c1 πs ∈ G π1 ∈ G X X π1 (a1 ) · · · πs (as )f (a1 , a2 , . . . , as ) ··· a1 ∈G1 as ∈Gs X X X X 1 = π1 (−x1 + a1 ) · · · πs (−xs + as )f (a1 , a2 , . . . , as ) |G1 ||G2 | · · · |Gs | a ∈G a ∈G 1 1 s 1 c1 c1 π1 ∈ G πs ∈ G 1 |G1 ||G2 | · · · |Gs |f (x1 , x2 , . . . , xs ) = |G1 ||G2 | · · · |Gs | = f (x1 , x2 , . . . , xs ), since X π(g) = b π∈G   |G| g = 0  0 g= 6 0. ✷ This is a generalization of the Fourier inversion formula. Next we generalize the Poisson summation formula. b : H) = {π ∈ G b | π| = 1} and note Before stating the next lemma we shall define (G H that   |H| π ∈ (G b : H) X π(x) = b : H)  0 π∈ / (G x∈H Lemma 7.2 Let Hi be a subgroup of Gi . For every ai in Gi we have X X x1 ∈H1 x2 ∈H2 ··· X f (a1 + x1 , a2 + x2 , . . . , as + xs ) = xs ∈Hs X 1 1 1 · · · c : H )| |(G c : H )| c : H )| |(G |(G 1 1 2 2 s s π ∈(G c :H 1 1 22 1) X c2 :H2 ) π2 ∈(G ··· X cs :Hs ) πs ∈(G fb(π1 , π2 , . . . , πs ). Proof. We have that X ··· x1 ∈H1 X f (a1 + x1 , . . . , as + xs ) xs ∈Hs X = x1 ∈H1 ··· X xs ∈Hs X X 1 π1 (−x1 ) · · · πs (−xs )fb(π1 , . . . , πs ) ··· |G1 | · · · |Gs | c c π1 ∈ G 1 X X 1 ··· |G1 | · · · |Gs | x1 ∈H1 xs ∈Hs = 1 |G1 | · · · |Gs | = 1 |G1 | · · · |Gs | = X ··· c1 π1 ∈ G X X cs πs ∈ G c1 :H1 ) π1 ∈(G ··· πs ∈ G s X ... c1 π1 ∈ G X cs πs ∈ G fb(π1 , . . . , πs ) X cs :Hs ) πs ∈(G π1 (−x1 ) · · · πs (−xs )fb(π1 , . . . , πs ) X π1 (−x1 ) x1 ∈H1 X πs (−xs ) xs ∈Hs fb(π1 , . . . , πs )|H1 | · · · |Hs | X X 1 · · · fb(π1 , . . . , πs ). c : H )| · · · |(G c : H )| |(G 1 1 s s π ∈(G c :H ) c :H ) π ∈(G = 1 1 1 s s s ✷ Lemma 7.3 Suppose f i : G1 × · · · × Gs → A, with A a complex algebra, are functions with m m i = 1, 2, . . . , m. Let F : Gm 1 × G2 × · · · × Gs → A be given by 1 m 1 m F (x11 , . . . , xm 1 , x2 , . . . , x 2 , . . . , x s , . . . , x s ) = m Y f i (xi1 , xi2 , . . . , xis ). i=1 Then Fb = Qm ci f, i=1 i.e. Fb (π11 , . . . , π1m , π21 , . . . , π2m , . . . , πs1 , . . . , πsm ) = Proof. Straightforward. Qm ci i i f (π , π , . . . , π i ). i=1 1 2 s ✷ Let Y1 = (a11 , . . . , an1 ) ∈ (R1 )n , Y2 = (a12 , . . . , an2 ) ∈ (R2 )n and Ys = (a1s , . . . , ans ) ∈ (Rs )n . Let f : R1 × R2 × · · · × Rs → A by f (a1 , a2 , . . . , as ) = Xa where a = (a1 , a2 , . . . , as ), then for 1 ≤ ij ≤ n define f i1 ,i2 ,...,is (ai1 , ai2 , . . . , ais ) and set Y F (Y1 , Y2 , . . . , Ys ) = f i1 ,i2 ,...,is . 1≤ij ≤|Rj | 2 ,...,is (π All that remains is for find f i1 ,id α1 , . . . , παs ) where πα (z) = χ([x, y]) and αi runs over the elements of Ri . 2 ,...,is (π f i1 ,id α1 , . . . , π αs ) = = X ... w1 ∈R1 X w1 ∈R1 ... s X Y ws ∈Rs i=1 s X Y ws ∈Rs i=1 23 παj (wi )f i1 ,i2 ,...,is (w1 , . . . , ws ) χi (αj wi )X(w1 ,...,ws ) Let a, b ∈ R, and define the matrix T by: Ta,b = i=s Y χai i bi , i=1 where ai , bi denote the i-th coordinate of a and b respectively and χi is the generating character of the ring Ri (such a character exists because the ring is Frobenius). Theorem 7.4 Let C1 , C2 , . . . , Cs be codes over the finite Frobenius rings R1 , R2 , . . . , Rs respectively. Then P (C1⊥ , C2⊥ , . . . , Cs⊥ )(Xa ) = 1 P (C1 , C2 , . . . , Cs )(T (Xa )), |C1 ||C2 | · · · |Cs | where T (Xa ) denotes the natural action of the matrix T . As a corollary to this theorem we get the MacWilliams relations for the joint, complete and Hamming weight enumerators. 7.2 The Weight Enumerator and the Chinese Remainder Theorem Let C = CRT (C1 , C2 , . . . , Cs ), assuming of course that the rings are such the Chinese Remainder Theorem applies. Then we have the following: WC (x0 , x1 , . . . , xk ) = P (C1 , C2 , . . . , Cs )(xcrt(a) ), where a ∈ R and crt(a) is the unique element given by the Chinese remainder theorem that is equivalent to ai in Ri . Notice that the matrix of Z2 × Z3 is not exactly the same as the matrix giving the MacWilliams relations for Z6 . However they are equivalent, in that it simply replaces one generating character with another. πi Note the generating character for Z6 is ω = e 3 . Hence the matrix, indexed by 0, 1, 2, 3, 4, 5, giving the MacWilliams relations are:  1 √ 6            1 1 1 1 1 1 1 ω ω2 −1 ω4 ω5 1 1 1 ω 2 −1 ω 4 ω4 1 ω2 1 −1 1 ω2 1 ω4 ω 4 −1 ω 2 24 1 ω5 ω4 −1 ω2 ω       .      2πi For the complete combined weight enumerator we have χF2 = −1 and χF3 = e 3 = ω 2 then χF2 χF3 = −ω 2 = ω 5 . Then the matrix, indexed by 00, 11, 02, 10, 01, 12, giving the MacWilliams relations is:  1 √ 6            1 1 1 1 1 1 1 ω5 ω4 −1 ω2 ω 1 1 1 ω 4 −1 ω 2 ω2 1 ω4 1 −1 1 ω4 1 ω2 ω 2 −1 ω 4 1 ω ω2 −1 ω4 ω5       .      This amounts to replacing the generating character χ(1) = ω with the generating character χ(1) = ω 5 . 7.3 Symmetrized Weight Enumerators Let Ui be a group of units of the ring Ri . We say two elements of Ri are equivalent, (denoted x ≈ y), if x = uy for some u ∈ Ri . Let Hi denote the set of equivalence classes in R generated by this equivalence relation and let H = H1 × H2 × · · · × Hs . Definition 2 The symmetrized combined weight enumerator is given by SU1 ,U2 ,...,Us (C1 , C2 , . . . , Cs )(X[a] ) = X Y n X[a][a] (c1 ,c2 ...,cs ) , (c1 ,c2 ,...,cs )∈C [a]∈H where n[a] = |{i | for j = 1 . . . s [cij ] = [ai ]}| and cij is the i-th coordinate of cj and ai is the i-th coordinate of a. Note that this is a generalization of the definition given in [23]. Let [a], [b] ∈ H, and define the matrix M by Ma,b = i=s Y X X χab i , i=1 a∈[ai ] b∈[bi ] where ai , bi denote the i-th coordinate of a and b respectively and χi is the generating character of the ring Ri (such a character exists because the ring is Frobenius). By specializing the variables in the previous lemmas we have the following. Theorem 7.5 The MacWilliams relations for the symmetrized weight enumerator are given by SU1 ,U2 ,...,Us (C1⊥ , C2⊥ , . . . , Cs⊥ )(X[a] ) = 1 SU ,U ,...,Us (C1 , C2 , . . . , Cs )(M (X[a] )), |C1 ||C2 | · · · |Cs | 1 2 where M (Xa ) denotes the natural action of the matrix M . 25 For the symmetrized weight enumerator as given in [23], the matrix giving the MacWilliams relations, indexed by 0, 1, 2, 3, is given by:   1  √   6   1 2 2 1  5 2 4 1 ω + ω ω + ω −1   , 2 4 4 2 1 ω +ω ω +ω 1   1 −2 2 −1 which is identical to the matrix M as given above if U1 = {1} and U2 = {1, 2}, except that the matrix is indexed by [0][0], [1][1], [0][1], [1][0]. The symmetrized weight enumerator of a self-dual code over Z6 is held invariant by this matrix as well as the matrix:   1 0 0 0    0 ω 0 0     .  0 0 ω4 0    0 0 0 ω3 The group of matrices holding the weight enumerator of a Type I code over Z6 is generated by these matrices. A Magma computation gives that the group has order 384 and the Molien series is given by t40 + 2 t36 + 2 t32 + 4 t28 + 3 t24 + 3 t20 + 4 t16 + 3 t12 + t8 + 1 , (1 + t4 )2 (t8 − t4 + 1) (t2 − t + 1)2 (t2 + t + 1)2 (t4 − t2 + 1)2 (t − 1)4 (t + 1)4 (t2 + 1)4 where the denominator is also (1 − t4 )(1 − t8 )(1 − t12 )(1 − t24 ). The Taylor series is 1 + t4 + 3t8 + 7t12 + 13t16 + 21t20 + 35t24 + · · · . Hence by inspection of the denominator there are 4 primary invariants and by inspection of the numerator 24 nontrivial secondary invariants. References [1] Bannai E., Dougherty S.T., Harada M. and Oura M., Type II codes, even unimodular lattices and invariant rings. preprint. [2] Bonnecaze A., Solé P. and Calderbank A.R., Quaternary quadratic residue codes and unimodular lattices. IEEE Trans. Inform. Theory 41 (1995), 366–377. [3] Bonnecaze A., Solé P., Bachoc C. and Mourrain B., Type II codes over Z4 . IEEE Trans. Inform. Theory 43 (1997), 969–976. 26 [4] Conway J.H., Pless V. and Sloane N.J.A., The binary self-dual codes of length up to 32: a revised enumeration. J. Combin. Theory Ser. A 60 (1992), 183–195. [5] Conway J.H. and Sloane N.J.A., On the enumeration of lattices of determinant one, J. Number Theory 15 (1982), 83–94. [6] Conway J.H. and Sloane N.J.A., Sphere Packings, Lattices and Groups. Springer-Verlag, New York, 1993. [7] Conway J.H. and Sloane N.J.A., Self-dual codes over the integers modulo 4. J. Combin. Theory Ser. A 62 (1993), 30–45. [8] Dougherty S.T., Gulliver T.A. and Harada M., Type II codes over finite rings and even unimodular lattices. J. Alg. Combin., (to appear). [9] Dougherty S.T. and Harada M., Self-dual codes constructed from Hadamard matrices and symmetric designs. (submitted). [10] Dougherty S.T., Harada M. and Solé P., Shadow codes over Z4 . Finite Fields and Their Appl., (submitted). [11] Hammons, Jr. A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A. and Solé P., A linear construction for certain Kerdock and Preparata codes. Bull Amer. Math. Soc. 29 (1993), 218–222. [12] Harada M., New extremal Type II codes over Z4 . Des. Codes and Cryptogr. 13 (1998), 271–284. [13] Harada M., Weighing matrices and self-dual codes. Ars Combin., 47 (1997), 65–73. [14] Huffman W.C., On extremal self-dual ternary codes of lengths 28 to 40. IEEE Trans. Inform. Theory 38 (1992), 1395–1400. [15] Klemm M., Über die Identät von MacWilliams für die Gewichtsfunktion von Codes. Archiv. Math, 49 (1987), 400–406. [16] Klemm M., Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4. Archiv. Math, 53 (1989), 201–207. [17] MacWilliams F.J. and Sloane N.J.A., The Theory of Error-Correcting Codes. NorthHolland, Amstterdam, 1977. [18] Nebe G., Private communication. 27 [19] Ohmori H., On the classifications of weighing matrices of order 12. J. Combin. Math. Combin. Comput. 5 (1989), 161–216. [20] Pless V., The number of isotropic subspaces in a finite geometry. Att. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 39 (1965), 418–421. [21] Pless V., On the uniqueness of the Golay codes. J. Combin. Theory 5 (1969), 215–228. [22] Pless V., Sloane N.J.A. and Ward H.N., Ternary codes of minimum weight 6 and the classification of the self-dual codes of length 20. IEEE Trans. Inform. Theory 26 (1980), 305–316. [23] Wood J., Duality for modules over finite rings and applications to coding theory. (submitted). 28