arXiv:1011.5382v2 [math.CO] 6 Apr 2011 On the Classification of Weighing Matrices and Self-Orthogonal Codes Masaaki Harada∗and Akihiro Munemasa† April 7, 2011 Abstract We provide a classification method of weighing matrices based on a classification of self-orthogonal codes. Using this method, we classify weighing matrices of orders up to 15 and order 17, by revising some known classification. In addition, we give a revised classification of weighing matrices of weight 5. A revised classification of ternary maximal self-orthogonal codes of lengths 18 and 19 is also presented. 1 Introduction A weighing matrix W of order n and weight k is an n × n (1, −1, 0)-matrix W such that W W T = kIn , where In is the identity matrix of order n and W T denotes the transpose of W . A weighing matrix of order n and weight n is also called a Hadamard matrix. We say that two weighing matrices W1 and W2 of order n and weight k are equivalent if there exist (1, −1, 0)-monomial matrices P and Q with W1 = P W2 Q. Chan, Rodger and Seberry [4] began a classification of weighing matrices and they classified all weighing matrices of weight k ≤ 5 and all weighing matrices of orders n ≤ 11. Ohmori [17] and [19] classified weighing matrices ∗ Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560, Japan, and PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332– 0012, Japan. email: mharada@sci.kj.yamagata-u.ac.jp † Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan. email: munemasa@math.is.tohoku.ac.jp 1 of orders 12 and 13, respectively. At order 14, weighing matrices of weights k ≤ 8 and 13 were classified in [4] and [18], respectively. At order 17, all weighing matrices of weight 9 with intersection number 8 were classified in [20]. In this paper, we extend the classification of weighing matrices using the known classification of self-orthogonal codes. Let Zm be the ring of integers modulo m, where m is an integer greater than 1. Let W be a weighing matrix of order n and weight k, and suppose that m is a divisor of k. If we regard the entries of W as elements of Zm , then the rows of W generate a self-orthogonal Zm -code. This means that W can be regarded as a subset of codewords in some maximal self-orthogonal code. For example, a classification of weighing matrices of order 16 and weight 6 can be derived from the known classification of ternary self-dual codes of length 16 given in [2]. The paper is organized as follows. In Section 2, we review the known classification of maximal self-orthogonal codes needed for our classification of weighing matrices. It turns out that there are errors in the classification of ternary maximal self-orthogonal codes of lengths 18 and 19 given in [22], and we correct them. In Section 3, we give a detailed description of our classification method of weighing matrices of order n and weight k based on the classification of self-orthogonal Zm -codes of length n, where m is a divisor of k. Our method, applied to the known classification of self-dual F5 -codes of length 12, leads to a classification of weighing matrices of order 12 and weight 5. This reveals an omission in the classification given in [4, Theorem 5], and a revised classification of weighing matrices of weight 5 for all orders is given in Section 4, while a revised classification of weighing matrices of order 12 for all weights is given in Section 5. In Section 6, we classify weighing matrices of orders 14, 15 and 17. Again, there is an error in the number of weighing matrices of order 14 and weight 8 given in [17, Theorem 3], and we correct it. This completes a classification of weighing matrices of orders n ≤ 17 except n = 16. Weighing matrices of order n and k are also classified for (n, k) = (16, 6), (16, 9), (16, 12), (18, 9) in Section 7. All weighing matrices given in this paper can be obtained electronically from [11]. 2 2 2.1 Maximal self-orthogonal codes Codes We shall exclusively deal with the case Zp = Fp and Z4 , where Fp denotes the finite field of odd prime order p. A Zm -code C of length n (or a code C of length n over Zm ) is a Zm -submodule of Znm . The dual code C ⊥ of C is defined as C ⊥ = {x ∈ Znm | x · y = 0 for all y ∈ C} under the standard inner product x · y. A code C is self-dual if C = C ⊥ , and C is self-orthogonal if C ⊂ C ⊥ . A self-dual Fp -code of length n exists if and only if n is even for p ≡ 1 (mod 4), and n ≡ 0 (mod 4) for p ≡ 3 (mod 4). A self-dual Z4 -code exists for every length. A self-orthogonal code C is maximal if C is the only self-orthogonal code containing C. The dimension of a maximal self-orthogonal Fp -code of length n is a constant depending only on n and p, and a self-dual code is automatically maximal. More precisely, for p ≡ 1 (mod 4), a maximal self-orthogonal Fp -code of length n has dimension (n − 1)/2 if n is odd. For p ≡ 3 (mod 4), a maximal self-orthogonal Fp -code of length n has dimension (n − 1)/2 if n is odd, n/2−1 if n ≡ 2 (mod 4). It is easy to see that a maximal self-orthogonal Z4 -code is necessarily self-dual for every length. Two codes C and C ′ are equivalent if there exists a (1, −1, 0)-monomial matrix P with C ′ = CP = {xP | x ∈ C}. The automorphism group Aut(C) of C is the group of all (1, −1, 0)-monomial matrices P with C = CP . Our classification method of weighing matrices of order n and weight k = mt requires a classification of maximal self-orthogonal Zm -codes of length n (see Section 3). In this paper, some classifications of maximal self-orthogonal Zm codes are used for m = 3, 4, 5, 7 to classify weighing matrices. The current knowledge on the classifications of such codes is listed in Table 1. 2.2 Ternary maximal self-orthogonal codes An F3 -code is called ternary. All ternary maximal self-orthogonal codes of lengths 4m + 1, 4m + 2, 4m + 3 can be obtained from self-dual codes of length 4m + 4 by subtracting (see [2]). A classification of ternary maximal self-orthogonal codes of lengths 3, . . . , 12, lengths 13, 14, 15, 16 and lengths 17, 18, 19, 20 was done in [15], [2] and [22], respectively. In the course of reproducing a classification of ternary maximal selforthogonal codes of lengths up to 20, we discovered errors in the classification 3 Table 1: Maximal self-orthogonal Zm -codes of length n Zm F3 Z4 F5 F7 Lengths n 1, . . . , 12 13, . . . , 16 17, . . . , 20 24 1, . . . , 9 10, . . . , 15 16 (Type II) 16 (Type I), 17, 18, 19 1, . . . , 12 13, . . . , 16 1, . . . , 9 10, . . . , 13 References [15] [2] [22] (see also this section) [9] [3] [5] [21] [10] [14] [8] [23] [7] for lengths 18 and 19. The numbers of ternary maximal self-orthogonal codes of lengths 18 and 19 are listed in [22, Table IV] as 154 and 54, respectively. However, we verified that the correct numbers are 160 and 56, respectively. Let C20,i denote the i-th self-dual code of length 20 given in [22, Tables II and III], and let n18 (i) and n19 (i) denote the numbers of inequivalent maximal self-orthogonal codes of lengths 18 and 19, respectively, obtained from C20,i (k) by subtracting. Let C20,i denote the self-orthogonal code of length 19 obtained from C20,i by subtracting the k-th coordinate. Although the numbers n19 (20) and n19 (23) are listed as both 1 in [22, Table IV], we verified that (i) the codes C20,20 (i = 1, . . . , 20) are equivalent to one of the two inequivalent (1) (20) (i) codes C20,20 , C20,20 , and the codes C20,23 (i = 1, . . . , 20) are equivalent to one (1) (20) of the two inequivalent codes C20,23 , C20,23 . In fact, these four codes have different automorphism groups, of orders 32, 128, 576 and 5184, respectively. Hence, we conclude that n19 (20) = n19 (23) = 2. Since [22, Table IV] also contains incorrect values for n18 (i), we list their correct values in Table 2. In order to check that a classification is complete, in all of the classification results, we first verified by Magma that all codes are inequivalent. This was done by the Magma function IsIsomorphic, as well as by checking that all codes have different numbers (B0 , B1 , . . . , Bn ), where Bj is the number of distinct cosets of weight j. Then we checked the mass formula, that is, we 4 computed the sum in X 2n · n! , | Aut(C)| C∈C (1) where C is the set of inequivalent maximal self-orthogonal codes of length n and we checked against the known formula for the number N0 of distinct maximal self-orthogonal codes of length n, which is given in [15, p. 650]. The automorphism group Aut(C) of C is calculated by the Magma function AutomorphismGroup. Note that each summand in (1) expresses the cardinality of the equivalence class of a given a code C and the sum of all these cardinalities is equal to N0 . The numbers # of all inequivalent maximal self-orthogonal codes of lengths up to 20 are listed in Table 3, and generator matrices of those codes can be obtained electronically from [11]. Proposition 1. Up to equivalence, there are 160 and 56 ternary maximal self-orthogonal codes of lengths 18 and 19, respectively. Table 2: Ternary maximal self-orthogonal codes of lengths 18 and 19 i 1 2 3 4 5 6 7 8 3 n18 (i) 2 5 4 7 7 6 4 5 n19 (i) 1 2 2 3 3 3 2 2 i 9 10 11 12 13 14 15 16 n18 (i) 5 8 4 12 8 9 10 7 n19 (i) i 2 17 2 18 2 19 4 20 3 21 3 22 3 23 2 24 Total n18 (i) 16 12 3 10 4 5 5 2 160 n19 (i) 5 4 1 2 1 1 2 1 56 Classification method When n is odd, the existence of a weighing matrix of order n and weight k implies that k is a square and (n − k)2 + (n − k) + 1 ≥ n. When n ≡ 2 (mod 4), the existence of a weighing matrix of order n and weight k implies that k is the sum of two squares and k ≤ n − 1 [4]. 5 Table 3: Ternary maximal self-orthogonal codes Length 3 4 5 6 7 8 9 10 11 # 1 1 1 2 1 1 2 5 3 References [15] [15] [15] [15] [15] [15] [15] [15] [15] Length 12 13 14 15 16 17 18 19 20 # 3 7 22 12 7 23 160 56 24 References [15] [2] [2] [2] [2] [22] Section 2.2 Section 2.2 [22] For the remainder of this section, P let W = (wij ) be a weighing matrix 2 2 of order n and weight k. The number ns=1 wis wjs is called the intersection number of i-th row ri and the j-th row rj (i 6= j). The maximum number among intersection numbers for rows of W and W T is called the intersection number of W [20]. We say that rj intersects ri in 2ℓ places if the intersection number is 2ℓ [4]. For a fixed row ri , let x2ℓ be the numbers of rows rj other than ri such that the intersection number of ri and rj is 2ℓ. The sequence (x0 , x2 , . . . , x2⌊n/2⌋ the intersection pattern corresponding to ri [4]. Pn ) is2called 2 2 The number j=1 wsj wtj wuj is called the generalized intersection number and the following set of generalized intersection numbers N(i) = n {s, t, u} | n X 2 2 2 wsj wtj wuj = i, 1 ≤ s, t, u ≤ n (s 6= t, s 6= u, t 6= u) j=1 o is called the g-distribution (see [20]). Note that there are inequivalent weighing matrices with the same g-distribution. Let Cm (W ) be the Zm -code generated by the rows of W , where the entries of W are regarded as elements of Zm . The following is trivial. Lemma 2. If k is divisible by m, then Cm (W ) is self-orthogonal. Proposition 3. Let p be an odd prime. If k is divisible by p but k is not divisible by p2 , then Cp (W ) is a self-dual Fp -code. 6 Proof. Suppose that k = pt, where t is not divisible by p. Since det(W 2 ) = det(W W T ) = det(kIn ) = k n , we have | det(W )| = k n/2 . Let d1 |d2 | · · · |dn be the elementary divisors of W (see e.g. [16, II.17] for the definition of elementary divisors). Then n n n | det(W )| = d1 d2 · · · dn = k 2 = p 2 t 2 . Since t is not divisible by p, n must be even, and at most n/2 di ’s are divisible by p. Hence, dim Cp (W ) ≥ n/2. By Lemma 2, dim Cp (W ) ≤ n/2. The result follows. From now on, suppose that Zm is either Fp or Z4 . Let ni (x) denote the number of components i of x ∈ Znm (i ∈ Zm ). Any row of W is a codeword x of Cm (W ) such that n0 (x) = n − k and n1 (x) + n−1 (x) = k. By Lemma 2, Cm (W ) is self-orthogonal. It follows that the rows of W are composed of n codewords x with n0 (x) = n − k and n1 (x) + n−1 (x) = k in some maximal self-orthogonal Zm -code of length n. We now describe how all weighing matrices of order n and weight k = mt can be constructed from maximal self-orthogonal Zm -codes of length n. Let C be a maximal self-orthogonal Zm -code of length n, and let V be the set of pairs {x, −x} satisfying the condition that n0 (x) = n−k, n1 (x)+n−1 (x) = k, x ∈ C. We define the simple undirected graph Γ, whose set of vertices is the set V and two vertices {x, −x}, {y, −y} ∈ V are adjacent if x y T = 0, where x = (x1 , . . . , xn ) ∈ {0, 1, −1}n ⊂ Zn is the vector with x mod m = x. It follows that the n-cliques in the graph Γ are precisely the set of weighing matrices which generate subcodes of C. It is clear that the group Aut(C) acts on the graph Γ as an automorphism group, and therefore, the classification of such weighing matrices reduces to finding a set of representatives of ncliques of Γ up to the action of Aut(C). This computation was performed in Magma [1], the results were then converted to weighing matrices. In this way, by considering all inequivalent maximal self-orthogonal Zm -codes of length n, we obtain a set of weighing matrices which contain a representative of every equivalence class of weighing matrices of order n and weight k = mt. Since a weighing matrix does not, in general generate a maximal selforthogonal code, two equivalent weighing matrices may be contained in two inequivalent maximal self-orthogonal codes. One could consider not only maximal but also all self-orthogonal codes, and then list only those weighing 7 matrices which generate the given code. This will avoid duplication of equivalent weighing matrices in the classification. However, we took a different approach for efficiency. Once we have a set of weighing matrices which could possibly contain equivalent pairs of weighing matrices, we perform equivalent testing by considering the associated incidence structures. This construction of incidence structures is given by [12, Theorem 6.8], and in our case, it is as follows. Given a weighing matrix W of order n, replacing 0, 1, −1 in each entry by the matrices       0 1 1 0 0 0 , , , 1 0 0 1 0 0 respectively, we obtain a (0, 1)-matrix of order 2n. This matrix defines a square incidence structure D(W ) with 2n points and 2n blocks. We may take the set of points of D(W ) to be P = {±1, ±2, . . . , ±n}, so that the permutation τ = (1, −1)(2, −2) · · · (n, −n) is a fixed-point-free involutive automorphism of D(W ). More precisely, the set of blocks B(W ) of D(W ) is B(W ) = {Biε | 1 ≤ i ≤ n, ε = ±1}, where Biε = {εwij j | 1 ≤ j ≤ n, wij 6= 0}. Here an automorphism of D(W ) is a permutation of P which maps B(W ) to B(W ). The set of all automorphisms is called the automorphism group and is denoted by Aut(D(W )). If we denote the orbits on P under τ by P1 , . . . , Pn , then the following conditions hold. (i) |B ∩ Pi | ≤ 1 for any i (1 ≤ i ≤ n) and any block B ∈ B(W ), (ii) for any two blocks B, B ′ ∈ B(W ) such that B ′ 6= B, B τ , |{i | B ∩ Pi = B ′ ∩ Pi 6= ∅}| = |{i | ∅ = 6 B ∩ Pi 6= B ′ ∩ Pi 6= ∅}|. Let W1 and W2 be weighing matrices of the same order and weight. We say that D(W1 ) and D(W2 ) are equivalent if there is a permutation σ of P which maps B(W1 ) to B(W2 ). Obviously, the equivalence of W1 and W2 implies that of D(W1 ) and D(W2 ). Conversely, the following lemma gives a criterion under which the equivalence of D(W1 ) and D(W2 ) implies that of W1 and W2 . 8 Lemma 4. Let W be a weighing matrix of order n, and let D(W ) be the square incidence structure defined by W . Suppose that τ = τ0 = (1, −1)(2, −2) · · · (n, −n) is the unique fixed-point-free involutive automorphism of D(W ) satisfying the conditions (i) and (ii) above, up to conjugacy in Aut(D(W )). If U is a weighing matrix such that D(U) is equivalent to D(W ), then U is equivalent to W . Proof. Let σ denote a map from D(U) to D(W ) giving an equivalence. This means that σ is a permutation of P which maps B(U) to B(W ). We first claim that τ = σ −1 τ0 σ satisfies the conditions (i) and (ii) above. Indeed, the orbits on P under τ are Pi = {i, −i}σ (1 ≤ i ≤ n). If B ∈ B(W ), −1 −1 then B σ ∈ B(U), hence |B ∩ Pi | = |B σ ∩ {i, −i}| ≤ 1. Thus, (i) holds. −1 −1 −1 −1 If B, B ′ ∈ B(W ) and B ′ 6= B, B τ , then B ′ σ 6= B σ and B ′ σ 6= B τ σ = −1 B σ τ0 . Since (ii) holds for B(U) and τ0 , we have |{i | B σ −1 ∩ {i, −i} = B ′ = |{i | ∅ = 6 Bσ −1 σ−1 ∩ {i, −i} = 6 ∅}| ∩ {i, −i} = 6 B′ σ−1 ∩ {i, −i} = 6 ∅}|. Thus, (ii) holds. Therefore, the claim is proved. By assumption, then, τ is conjugate to τ0 in Aut(D(W )). This implies that there exists an automorphism π ∈ Aut(D(W )) such that σ −1 τ0 σ = π −1 τ0 π. Replacing σ by σπ −1 , we may assume from the beginning that σ commutes with τ0 . Then there exists a permutation ρ ∈ Sn and qj ∈ {±1} such that (±j)σ = ±qj j ρ . Let B(W ) = {Biε | 1 ≤ i ≤ n, ε = ±1}, B(U) = {Ciε | 1 ≤ i ≤ n, ε = ±1}, where Biε = {εwij j | 1 ≤ j ≤ n, wij 6= 0}, Ciε = {εuij j | 1 ≤ j ≤ n, uij = 6 0}. Since B(U)σ = B(W ), for any i, there exists i′ and and pi ∈ {±1} such i that (Ci+ )σ = Bip′i . Since σ commutes with τ0 , we have (Ci− )σ = Bi−p . This ′ implies that there exists a permutation π ∈ Sn such that (Ci+ )σ = Bipπi . Thus, qj uij = pi wiπ ,j ρ . Now, define monomial matrices P = (pi δiπ ,j ), Q = (qi δiρ ,j ). Then we obtain P W Q−1 = U. Therefore, W is equivalent to U. 9 4 Weighing matrices of weight 5 In the course of reproducing a classification of weighing matrices of order 12 (see Section 5), we discovered errors in the classification of weighing matrices of weight 5 given in [4, Theorem 5]. In this section, we give a revised classification of weighing matrices of weight 5. In the proof of [4, Theorem 5], the authors of [4] divide the classification into the following three cases: (a) at least two other rows intersect the first row in four places or, (b) no rows intersect any other row in four places or, (c) exactly one row intersects the first row in four places. Then all weighing matrices of weight 5 for the three cases (a), (b) and (c) were classified in [4, Theorem 5]. In the proof of [4, Theorem 5], D(16, 5) is claimed to be the unique weighing matrix of weight 5 satisfying (b). However, we found more weighing matrices of weight 5 satisfying (b). In Figure 1, we give such weighing matrices W12,5 and W14,5 of orders 12 and 14, respectively. Lemma 5. Let W be a weighing matrix of order 2n and weight 5 satisfying the condition (b). Then W contains W12,5 , W14,5 or D(16, 5) as a direct summand. Proof. From the condition (b), we may assume without that the first 5 rows of W have the following form:  + + + + + 0 0 0 0 0 0  + − 0 0 0 + + + 0 0 0  M1 =   + 0 − 0 0 − 0 0 + + 0  + 0 0 − 0 0 − 0 − 0 + + 0 0 0 − 0 0 − 0 − − loss of generality 0···0 0···0 0···0 0···0 0···0    ,   where +, − denote 1, −1, respectively. In addition, we may assume without loss of generality that the next three rows have the following form:   0 + − 0 0 M2 =  0 + 0 − 0 A B C , 0 + 0 0 − 10 W12,5 W14,5  1 1 1 −1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0  1 1 1 −1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0          =                     =             1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0   −1 0 0 −1 0 0 1 1 0 0   0 −1 0 0 −1 0 −1 0 1 0   0 0 −1 0 0 −1 0 −1 −1 0   −1 0 0 0 0 1 0 −1 0 1   0 −1 0 1 0 0 0 1 −1 0   0 0 −1 0 1 0 0 0 1 −1   1 −1 0 −1 1 0 0 0 0 1   1 0 −1 0 −1 1 1 0 0 0   0 1 −1 0 0 0 −1 1 0 1  0 0 0 1 0 −1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 −1 0 0 −1 0 0 1 1 0 0 0 −1 0 0 −1 0 −1 0 1 0 0 0 −1 0 0 −1 0 −1 −1 0 −1 0 0 1 0 0 0 0 0 1 0 −1 0 0 0 1 0 0 −1 0 0 0 −1 0 1 0 0 0 1 −1 1 −1 0 0 0 0 1 0 0 0 1 0 −1 0 0 0 0 1 0 1 0 1 −1 0 −1 1 0 0 0 0 0 0 0 1 −1 0 1 0 0 −1 0 0 0 1 0 −1 0 1 0 0 0 0 0 0 0 0 1 −1 1 1  0 0 0 0   0 0   0 0   0 0   1 0   −1 0   0 0   1 −1   0 1   0 −1   0 1   −1 −1  −1 0 Figure 1: Weighing matrices of orders 12, 14 and weight 5 where A is a 3 × 3 permutation matrix, B is some 3 × 3 matrix and C is some 3 × (2n − 11) matrix. Let M(A, B, C) denote the matrix   M1 . M2 If A′ = P1 AP1−1 for some 3 × 3 permutation matrix P1 , then P M(A, B, C)P −1 = M(A′ , B, C), 11 where   I2  P =  P1 P1 I2n−8  .  This means that it is sufficient to consider the matrix A up to conjugacy in the symmetric group of degree 3, so we assume A = I3 , A2 or A3 , where     + 0 0 0 0 + A2 =  0 0 +  and A3 =  + 0 0  . 0 + 0 0 + 0 • Case A = I3 : From the orthogonality of rows,     0 0 0 + + 0 0···0 B =  0 0 0  and C =  − 0 − 0 · · · 0  . 0 0 0 0 − + 0···0 Moreover, the matrix M(A, B, C) is uniquely extended to    W =  D(16, 5) O O  ,  ∗ up to equivalence, where O is the zero matrix. • Case A = A2 : From the orthogonality  0 0 B= 0 0 0 0 of rows,    0 + + 0···0 −  and C =  0 − 0 · · · 0  . + − 0 0···0 Moreover, the matrix M(A, B, C) is uniquely extended to    W =  W14,5 O O  ,  ∗ up to equivalence, where W14,5 is given in Figure 1. 12 • Case A = A3 : From the orthogonality of rows, B must be one matrices:      0 − 0 − 0 −  0 + −  ,  + 0 0  and  0 0 + 0 0 + of the following three  0 − 0 + 0 0 . − + 0 Then C can be considered as       + 0···0 0 0···0 + 0···0  0 0 · · · 0  ,  + 0 · · · 0  and  − 0 · · · 0  , − 0···0 − 0···0 0 0···0 respectively. Moreover, for each case the matrix M(A, B, C) is uniquely extended to    W =  W12,5 O O  ,  ∗ up to equivalence, where W12,5 is given in Figure 1. Therefore, W contains W12,5 , W14,5 or D(16, 5) as a direct summand. Remark 6. For order 14, it follows from [4, Theorem 5] that there are two inequivalent weighing matrices of weight 5, namely, E(14, 5) and W (6, 5) ⊕ W (8, 5) in [4]. On the other hand, the table in [4, Appendix B] lists the number of inequivalent weighing matrices of weight 5 to be three, and the missing matrix is denoted by D(14, 5) which, however, is not defined in [4]. Remark 7. Let R = (rij ) be the square matrix of order n with rij = 1 if i + j − 1 = n and 0 otherwise. If A1 and A2 are circulant matrices of order n with entries 0, ±1 satisfying A1 AT1 + A2 AT2 = kI, then the matrices     A1 A2 A1 A2 R W1 = and W2 = −AT2 AT1 −A2 R A1 are weighing matrices of order 2n and weight k [6, Proposition 4.46]. Kotsireas and Koukouvinos [13] claim that all weighing matrices of the form W1 or W2 are found by an exhaustive search for n ≤ 11. Although the results of 13 their search are not given, this means that they must have found the weighing matrix W14,5 , since it is equivalent to the weighing matrix W1 where A1 and A2 are the circulant matrices with first rows (1, 0, 0, 0, 0, 0, 0) and (−1, 1, 1, 0, 1, 0, 0), respectively. We verified that no weighing matrices W1 , W2 constructed from two circulant matrices A1 and A2 are equivalent to W12,5 . This was done by finding all weighing matrices of the form W1 and W2 by an exhaustive search. Remark 8. Let W be any of W12,5 , W14,5 and D(16, 5). Then W T also satisfies (b). Let W be the (1, 0)-matrix obtained from W by changing −1 to 1 in the entries. Then W is the incidence matrix of a semibiplane (see [25] for the definition of a semibiplane). The three semibiplanes obtained in this way are given in [25, Proposition 15]. By the above lemma, we have the following revised classification for weight 5. See [4] for the definitions of the weighing matrices W (6, 5), W (8, 5), E(4ti + 2, 5) and F (4tj + 4, 5). Theorem 9. Any weighing matrix of order 2n and weight 5 is equivalent to M i1 W (6, 5) M W (8, 5) W12,5 D(16, 5) i5 M M M ti W14,5 i4 i3 i2 M M E(4ti + 2, 5)
M M tj i6 i8
F (4tj + 4, 5) , where ti ≥ 2 and tj ≥ 2. Table 4 is a revised table of a classification of weighing matrices of order 2n ≤ 20 and weight 5 in [4, Appendix B]. 5 Weighing matrices of order 12 The classification of weighing matrices of order 12 was done in [4] and [17]. In this section, we give a revised list of weighing matrices of weights 6, 8, 10. These classifications were done by considering self-dual Zk -codes of length 12, where k = 3, 4 and 5, respectively, using the method in Section 3. These approaches are similar, and we give details only for weight 6. 14 Table 4: Weighing matrices of weight 5 2n 6 8 10 12 14 16 18 # 1 1 1 3 3 4 5 20 7 5.1 Matrices W (6, 5) W (8, 5) E(10, 5) W12,5 , F (12, 5), W (6, 5) ⊕ W (6, 5) W14,5 , E(14, 5), W (6, 5) ⊕ W (8, 5) D(16, 5), F (16, 5), W (8, 5) ⊕ W (8, 5), W (6, 5) ⊕ E(10, 5) E(18, 5), W (6, 5) ⊕ W12,5 , W (6, 5) ⊕ F (12, 5), W (6, 5) ⊕ W (6, 5) ⊕ W (6, 5) W (8, 5) ⊕ F (10, 5) F (20, 5), W (6, 5) ⊕ W14,5 , W (6, 5) ⊕ E(14, 5), W (6, 5) ⊕ W (6, 5) ⊕ W (8, 5) W (8, 5) ⊕ W12,5 , W (8, 5) ⊕ F (12, 5), E(10, 5) ⊕ E(10, 5) Weight 6 As described in Section 3, any weighing matrix of order 12 and weight 6 can be regarded as 12 codewords of weight 6 in some ternary self-dual code of length 12. There are three inequivalent ternary self-dual codes of length 12 [15, Table 1], and these codes are denoted by G12 , 4C3 (12) and 3E4 . The code G12 has minimum weight 6 and the other codes have minimum weight 3, and the numbers of codewords of weight 6 in these codes are 264, 240 and 192, respectively. By considering sets of 12 codewords of weight 6 in these codes, we have the following classification of weighing matrices of order 12 and weight 6, using the method in Section 3. Theorem 10. There are 8 inequivalent weighing matrices of order 12 and weight 6. The number of inequivalent weighing matrices of order 12 and weight 6 was incorrectly reported as 7 in [17, Theorem 5]. The 7 inequivalent matrices ∗ ∗ T in [17, Theorem 5] are denoted by E1∗ , E2∗ , (E2∗ )T , E5∗ , E14 , (E14 ) and G∗2 . The missing matrix W12,6 is listed in Figure 2. We remark that W12,6 and T W12,6 are equivalent. Remark 11. It is claimed in the proof of [17, Lemma 31] that there are 4 weighing matrices which are constructed from Case II up to equivalence. The matrix W12,6 is also constructed from Case II, and hence there are 5 weighing matrices which are constructed from Case II up to equivalence. 15 W12,6 W12,10  1 1 1 1 1 1 −1 0 1 −1 0 1 1 −1 0 0 1 0 0 −1 1 0 0 −1 0 1 1 0 0 1 −1 0 0 0 1 −1 0 0 1 0 0 0 0 1 0 0 0 0  1 1 0 −1 −1 1 1 0 −1 1 −1 1          =                   =          1 1 1 0 −1 −1 1 1 0 −1 1 −1 1 1 1 1 0 −1 −1 −1 1 0 −1 1 1 1 −1 1 1 0 −1 1 −1 1 0 −1 1 −1 −1 0 1 0 −1 0 −1 0 0 0 1 0 0 0 −1 0 −1 0 1 −1 −1 0 1 1 −1 −1 1 1 0 −1 1 −1 1 0 1 −1 0 −1 1 −1 1 0 1 1 −1 −1 0 0 0 0 0 0 1 −1 −1 −1 −1 1 1 −1 1 0 −1 1 −1 −1 0 1 1 −1 0 0 0 0 0 1 1 0 0 0 −1 0 0 0 1 1 −1 0 1 −1 0 0 1 0 1 −1 0 −1 −1 −1 0 −1 0 0 0 −1 −1 0 1 0 0 0 1 0 0 0 1 −1 1 0 0 0 1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 0 −1 1 −1 −1 0 1 1 1 −1 1 −1 1 0 −1 1 −1 −1 0 1 1 −1 −1 1 −1 1 0 1 1 −1 −1 0 0 0 1 1 1 1 1 −1 −1 −1 −1 −1 0 0 1 1 1 1 1 1 1 1 1 1                                         Figure 2: Weighing matrices W12,6 and W12,10 In Table 5, we list g-distributions N(i) (i = 0, 1, . . . , 6) for the 7 matrices given in [17, Theorem 5] along with the new matrix W12,6 . Table 5 also shows that the 8 weighing matrices are inequivalent. By Proposition 3, the ternary codes C3 (W ) generated by the rows of these matrices W are self-dual, and the identifications with those appearing in [15] are given in the last column of Table 5. 5.2 Weight 8 According to [17, Theorem 3], there are 6 inequivalent weighing matrices of order 12 and weight 8. However, our method in Section 3, applied to the known classification of self-dual Z4 -codes of length 12 given in [5], leads to 16 Table 5: Weighing matrices of order 12 and weight 6 W E1∗ E2∗ (E2∗ )T E5∗ ∗ E14 ∗ )T (E14 G∗2 W12,6 N (0) 396 516 432 492 708 384 432 516 N (1) 720 432 528 432 0 576 576 360 N (2) 180 420 504 468 756 576 456 540 N (3) 240 144 48 144 0 0 0 120 N (4) 180 204 192 180 252 144 240 180 N (5) 0 0 0 0 0 0 0 0 N (6) 0 0 12 0 0 36 12 0 C3 (W ) G12 G12 4C3 (12) G12 G12 3E4 4C3 (12) G12 the following classification of weighing matrices of order 12 and weight 8. Theorem 12. There are 7 inequivalent weighing matrices of order 12 and weight 8. Remark 13. The 6 inequivalent matrices in [17, Theorem 3] are denoted by A1 , A3 , A6 , A7 , A8 and A9 . However, A11 appeared in the proof of [17, Theorem 3] is inequivalent to any of the matrices Ai (i = 1, 3, 6, 7, 8, 9). This is an error in [17, Theorem 3]. Let W be a weighing matrix of order 12 and weight 8. Let D4 (W ) be the Z4 -code with generator matrix ( I12 , W ), where the matrix W is regarded as a matrix over Z4 . The numbers #D6 of codewords of weight 6 of D4 (W ), listed in Table 6, were found by the Magma function NumberOfWords. These numbers also show that the 7 weighing matrices are inequivalent, Table 6: Weighing matrices of order 12 and weight 8 W #D6 5.3 A1 2852 A3 1764 A6 1092 A7 932 A8 1124 A9 1700 A11 1220 Weight 10 According to [17, Theorem 1], there are 4 inequivalent weighing matrices of order 12 and weight 10. However, our method in Section 3, applied to the 17 known classification of self-dual F5 -codes of length 12 given in [14], leads to the following classification of weighing matrices of order 12 and weight 10. Theorem 14. There are 5 inequivalent weighing matrices of order 12 and weight 10. The 4 inequivalent matrices in [17, Theorem 1] are denoted by A1 , A4 , A7 and A8 . The missing matrix W12,10 is listed in Figure 2. We remark that T W12,10 and W12,10 are equivalent. Remark 15. It is claimed in the proof of [17, Lemma 11] that there are only 7 vectors such that the matrices Yi are normal matrices of level 4. We verified that this is incorrect and there is one missing vector, namely, the fourth row of W12,10 . Moreover, the 7 × 12 matrix Ȳ4 consisting of the first 7 rows of W12,10 should be considered in [17, Lemma 12] as a possible matrix of level 7. In Table 7, we list the self-dual F5 -codes C5 (W ) generated by the rows of these matrices W , in the notation of [14]. This shows that W12,6 must be inequivalent to any of the other 4 matrices. We consider F5 -codes D5 (W ) with generator matrices ( I12 , W ), where the matrices W are regarded as matrices over F5 . The numbers #D8 of codewords of weight 8 are listed in Table 7, which also shows that the 5 weighing matrices are inequivalent. These numbers were found by the Magma function NumberOfWords. Table 7: Weighing matrices of order 12 and weight 10 W A1 A4 A7 A8 W12,10 5.4 C5 (W ) F62 F12 F12 F62 K12 #D8 3696 3000 4080 4560 3792 Other weights By Theorem 9 (see Table 4), there are 3 inequivalent weighing matrices of order 12 and weight 5, namely, W12,5 , F (12, 5), and W (6, 5) ⊕ W (6, 5). In 18 Table 8, we list the self-dual F5 -codes C5 (W ) generated by the rows of these matrices W , in the notation of [14]. Table 8: Weighing matrices of order 12 and weight 5 W C5 (W ) W (6, 5) ⊕ W (6, 5) F62 F (12, 5) F12 W12,5 K12 For weights 7 and 9, we verified that the classifications in [17] are correct, using the classification of self-dual Fp -codes of length 12, where p = 7 and 3, respectively. Table 9 summarizes a revised classification of weighing matrices of order 12. Table 9: Classification of weighing matrices of order 12 Weight 1 2 3 4 5 6 6 # 1 1 1 5 3 8 References [4] [4] [4] [4] Theorem 9 Theorem 10 Weight 7 8 9 10 11 12 # 3 7 4 5 1 1 References [17] Theorem 12 [17] Theorem 14 [4] [24] Weighing matrices of orders 14, 15 and 17 We continue a classification of weighing matrices using the method in Section 3. Then we have the following classification of weighing matrices of order n and weight k for (n, k) = (14, 8), (14, 9), (14, 10), (15, 9) and (17, 9), (2) using the classification of maximal self-orthogonal Zm -codes of length n (see Section 2), where m = 4, 3, 5, 3 and 3, respectively. Since approaches are similar to that used in Section 5, we only list in Table 10 the numbers # of inequivalent weighing matrices of order n and weight k for (n, k) listed in (2). 19 Hence, our result completes a classification of weighing matrices of orders up to 15 and order 17. Table 10: Classification of weighing matrices of orders 14, 15 and 17 Order 14 Weight 1 2 4 5 8 9 10 13 # 1 1 3 3 66 7 19 1 References [4] [4] [4] Theorem 9 Section 6 Section 6 Section 6 [4] Order 15 17 Weight 1 4 9 1 4 9 16 # 1 6 37 1 3 2360 1 References [4] [4] Section 6 [4] [4] Section 6 [4] Now, we compare our classifications with the known classifications for (n, k) = (14, 8) and (17, 9). According to [18, Theorem 3.9], there are 65 inequivalent weighing matrices of order 14 and weight 8. Using the classification of self-dual Z4 -codes of length 14, we classified weighing matrices of order 14 and weight 8, and we claim that the classification in [18, Theorem 3.9] misses the matrix W14,8 , which is listed in Figure 3. We remark that T W14,8 and W14,8 are equivalent. Hence, we have the following: Theorem 16. There are 66 inequivalent weighing matrices of order 14 and weight 8. T Remark 17. The intersection patterns of W14,8 and W14,8 are (x2 , x4 , x6 , x8 ) = (0, 11, 2, 0) which is the same as c25 in [18, p. 139]. Hence, W14,8 is of Type c25 in the sense of [18]. It is claimed in [18, Theorem 3.6] that a matrix of Type c25 is equivalent to some matrix of Type ci (i 6= 25). This is an error. Among the weighing matrices of order 17 and weight 9, Ohmori and Miyamoto [20] claimed to classify those with intersection number 8, and they found exactly 925 such matrices. However, we verified that only 517 of the 2360 weighing matrices of order 17 and weight 9 have intersection number 8. Since their list of 925 weighing matrices is not available, we are unable to compare their result with ours. 20  W14,8            =            1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 −1 1 1 −1 −1 −1 0 0 0 0 1 0 −1 0 0 0 −1 1 0 0 0 −1 0 0 1 0 0 −1 1 −1 0 0 −1 −1 0 1 1 0 0 0 1 −1 0 0 −1 1 0 1 −1 0 1 0 0 −1 −1 0 1 −1 −1 1 0 1 0 1 −1 1 0 1 0 −1 0 −1 0 −1 1 −1 1 0 0 1 0 −1 −1 0 −1 1 1 −1 0 0 0 0 −1 0 1 −1 1 0 0 0 −1 −1 1 0 −1 1 0 0 1 −1 0 −1 1 0 0 1 0 −1 0 −1 −1  0 0 0 0 0 0 0 0   1 1 0 0   1 −1 1 0   −1 0 1 1   −1 0 −1 −1   0 1 0 −1  . 0 −1 −1 1   0 1 0 1   0 1 −1 1   1 0 −1 −1   −1 0 1 −1   −1 1 0 0  −1 −1 −1 0 Figure 3: Weighing matrix W14,8 7 Other orders and weights For orders n ≥ 13 and weights k ≥ 6, we classified weighing matrices of some orders n and weights k listed in Table 11 using the classification of maximal self-orthogonal Fp -codes of length n given in Table 1. Since approaches are similar to that used in Section 5, we only list in Table 11 the numbers # of inequivalent weighing matrices for which we classified, and the primes p. Also, we list in the same table the orders and weights for which we checked the known classifications by our classification method, along with references. Table 11: Other orders and weights (n, k) (13, 9) (16, 6) (16, 9) (16, 10) (16, 12) # 8 30 704 670 279 p 3 3 3 5 3 References [19] (n, k) (16, 15) (18, 9) (20, 6) (20, 18) (24, 6) 21 # 1 11891 49 53 190 p 3 3 3 3 3 References [4] References [1] W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydney, Available online at http://magma.maths.usyd.edu.au/magma/. [2] J.H. Conway, V. Pless and N.J.A. Sloane, Self-dual codes over GF (3) and GF (4) of length not exceeding 16, IEEE Trans. Inform. Theory 25 (1979), 312–322. [3] J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A 62 (1993), 30–45. [4] H.C. Chan, C.A. Rodger and J. Seberry, On inequivalent weighing matrices, Ars Combin. 21 (1986), 299–333. [5] J. Fields, P. Gaborit, J.S. Leon and V. Pless, All self-dual Z4 codes of length 15 or less are known, IEEE Trans. Inform. Theory 44 (1998), 311–322. [6] A.V. Geramita and J. Seberry, Orthogonal Designs, Quadratic Forms and Hadamard Matrices, Lecture Notes in Pure and Applied Mathematics, 45, Marcel Dekker Inc., New York, 1979. [7] M. Harada and P.R.J. Östergård, Self-dual and maximal self-orthogonal codes over F7 , Discrete Math. 256 (2002), 471–477. [8] M. Harada and P.R.J. Östergård, On the classification of self-dual codes over F5 , Graphs Combin. 19 (2003), 203–214. [9] M. Harada and A. Munemasa, A complete classification of ternary selfdual codes of length 24, J. Combin. Theory Ser. A, 116 (2009), 1063– 1072. [10] M. Harada and A. Munemasa, On the classification of self-dual Zk -codes, Lecture Notes in Comput. Sci. 5921 (2009), 78–90. [11] M. Harada and A. Munemasa, Database of Self-Dual Codes, Available online at http://www.math.is.tohoku.ac.jp/~munemasa/ selfdualcodes.htm. 22 [12] D. Jungnickel, On automorphism groups of divisible designs, Canad. J. Math. 34 (1982), 257–297. [13] I.S. Kotsireas and C. Koukouvinos, New weighing matrices of order 2n and weight 2n − 5, J. Combin. Math. Combin. Comput. 70 (2009), 197– 205. [14] J.S. Leon, V. Pless and N.J.A. Sloane, Self-dual codes over GF(5), J. Combin. Theory Ser. A 32 (1982), 178–194. [15] C.L. Mallows, V. Pless and N.J.A. Sloane, Self-dual codes over GF(3), SIAM J. Appl. Math. 31 (1976), 649–666. [16] M. Newman, Integral matrices, Pure and Applied Mathematics 45, Academic Press, New York, 1972. [17] H. Ohmori, On the classifications of weighing matrices of order 12, J. Combin. Math. Combin. Comput. 5 (1989), 161–215. [18] H. Ohmori, Classification of weighing matrices of small orders, Hiroshima Math. J. 22 (1992), 129–176. [19] H. Ohmori, Classification of weighing matrices of order 13 and weight 9, Discrete Math. 116 (1993), 55–78. [20] H. Ohmori and T. Miyamoto, Construction of weighing matrices W (17, 9) having the intersection number 8, Des. Codes Cryptogr. 15 (1998), 259–269. [21] V. Pless, J.S. Leon and J. Fields, All Z4 codes of Type II and length 16 are known, J. Combin. Theory Ser. A 78 (1997), 32–50. [22] V. Pless, N.J.A. Sloane and H.N. Ward, Ternary codes of minimum weight 6 and the classification of length 20, IEEE Trans. Inform. Theory 26 (1980), 305–316. [23] V.S. Pless and V.D. Tonchev, Self-dual codes over GF (7), IEEE Trans. Inform. Theory 33 (1987), 723–727. [24] J.A. Todd, A combinatorial problem, J. Math. Phys. 12 (1933), 321–333. [25] P. Wild, Generalized Hussain graphs and semibiplanes with k ≤ 6, Ars Combin. 14 (1982), 147–167. 23