Orthogonal Designs of Order 32 and 64 via Computational Algebra

Baumert and Hall describe a Williamson array construction based on quaternions. We extend by analogy this construction to larger arrays, using the multiplication table of the Cayley-Dickson algebras of dimensions 32 and 64. Then we use Gröbner bases to obtain full orthogonal designs of order 32 with 10 variables and of order 64 in 10 and 11 variables. Finally we use OD (32; 1, 1, 2, 4, 4, 4, 4, 4, 4, 4) to search for inequivalent Hadamard matrices of order 96, 160, 224, 288. Such structured matrices are needed in Statistics and Coding Theory applications. This algebraic approach can be extended to larger orders, i.e. 2n, n≥7, provided that the structural properties of the corresponding polynomial ideals and their Gröbner bases are further investigated and understood

Genetic Algorithm for Orthogonal Designs

We show how to use Simple Genetic Algorithm to produce Hadamard matrices of large orders, from teh full orthogonal design or oder 16 with 9 variables, OD(16; 1, 1, 2, 2, 2, 2, 2, 2, 2). The objective functionthat we use in our implementation of Simple Genetic Algorithm, comes from a Computational Algebra formalism of the full orthogonal design equations. In particular, we constructed Hadamard matrices of orders 144, 176, 208, 240, 272, 304 and 336, from the aforementioned orthogonal design. By varying three genetic operator parameters, we computer 62 inequivalent Hadamard matices of order 304 and 4 inequivalent Hadamard matrices of order 336. Therefore we established two new constructive lower bounds for the numbers of Hadamard matrices of order 304 and 336

Orthogonal Designs of Order 32 and 64 via Computational Algebra

Baumert and Hall describe a Williamson array construction based on quaternions. We extend by analogy this construction to larger arrays, using the multiplication table of the Cayley-Dickson algebras of dimensions 32 and 64. Then we use Gröbner bases to obtain full orthogonal designs of order 32 with 10 variables and of order 64 in 10 and 11 variables. Finally we use OD (32; 1, 1, 2, 4, 4, 4, 4, 4, 4, 4) to search for inequivalent Hadamard matrices of order 96, 160, 224, 288. Such structured matrices are needed in Statistics and Coding Theory applications. This algebraic approach can be extended to larger orders, i.e. 2n, n≥7, provided that the structural properties of the corresponding polynomial ideals and their Gröbner bases are further investigated and understood

On the complete pivoting conjecture for Hadamard matrices of small orders

In this paper we study explicitly the pivot structure of Hadamard matrices of small orders 16, 20 and 32. An algorithm computing the (n — j) x (n — j) minors of Hadamard matrices is presented and its implementation for n = 12 is described. Analytical tables summarizing the pivot patterns attained are given

Numerical algorithms for the computation of the Smith normal form of integral matrices,

Numerical algorithms for the computation of the Smith normal form of integral matrices are described. More specifically, the compound matrix method, methods based on elementary row or column operations and methods using modular or p-adic arithmetic are presented. A variety of examples and numerical results are given illustrating the execution of the algorithms

Necessary and sufficient conditions for two variable orthogonal designs in order 44: Addendum

In our recent paper Necessary and sufficient conditions for some two variable orthogonal designs in order 44, Koukouvinos, Mitrouli and Seberry leave 7 cases unresolved. Using a new algorithm given in our paper A new algorithm for computer searches for orthogonal designs by the present four authors we are able to finally resolve all these cases. This note records that the necessary conditions for the existence of two variable designs constructed using four circulant matrices are sufficient. In particular of 484 potential cases 404 cases have been found, 68 cases do not exist and 12 cases cannot be constructed using four circulant matrices

Necessary and Sufficient Conditions for Three and Four Variable Orthogonal Designs in Order 36

We use a new algorithm to find new sets of sequences with entries from {0, ±a, ±b, ±c, ±d}, on the commuting variables a, b, c, d, with zero autocorrelation function. Then we use these sequences to construct a series of new three and four variable orthogonal designs in order 36. We show that the necessary conditions plus (.s1, s2, s3, s4) not equal to 12816 18 816 221313 26721 36 816 4889 12825 191313 23 424 289 9 381015 8899 14425 22 916 are sufficient for the existence of an OD(36; s1, s2 s3, s4) constructed using four circulant matrices in the Goethals-Seidel array. Of the 154 theoretically possible cases 133 are known. We also show that the necessary conditions plus (s1, s2 s3) ≠ (2,8,25), (6,7,21), (8, 9, 17) or (9, 13, 13) are sufficient for the existence of an OD(36; s1, s2, s3) constructed using four circulant matrices in the Goethals-Seidel array. Of the 433 theoretically possible cases 429 are known. Further, we show that the necessary conditions are sufficient for the existence of an OD(36; s1, s2 36 — s1 — s2) in each of the 54 theoretically possible cases. Further, of the 27 theoretically possible OD(36; s1, s2, s3, 36 — s1 — s2 — s3), 23 are known to exist, and four, (1,2,8,25), (1, 9,13,13), (2,6,7,21) and (3, 8,10,15), cannot be constructed using four circulant matrices. By suitably replacing the variables by ±1 these lead to more than 200 potentially inequivalent Hadamard matrices of order 36. By considering the 12 OD(36; 1, s1, 35—s1) and suitably replacing the variables by ±1 we obtain 48 potentially inequivalent skew-Hadamard matrices of order 36. A summary with all known results in order 36 is presented in the Tables

On sufficient conditions for some orthogonal designs and sequences with zero autocorrelation function

We give new sets of sequences with entries from {0, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d and zero autocorrelation function. Then we use these sequences to construct some new orthogonal designs. This means that for order 28 only the existence of the following five cases, none of which is ruled out by known theoretical results, remain in doubt: OD(28; 1, 4, 9, 9), OD(28; 1, 8, 8, 9), OD(28; 2, 8, 9, 9), OD(28; 3, 6, 8, 9), OD(28; 4, 4, 4, 9). We consider 4 - N PAF(Sl, S2, S3, S4) sequences or four sequences of commuting variables from the set {0, ±a, ±b, ±c, ±d} with zero nonperiodic autocorrelation function where ±a occurs Sl times, ±b occurs S2 times, etc. We show the necessary conditions for the existence of an 0D(4n; S1,S2, S3,S4) constructed using four circulant matrices are sufficient conditions for the existence of 4 - NPAF(S1, S2, S3, S4) sequences for all lengths ≥ n, i) for n = 3, with the extra condition (S1,S2,S3,S4) ≠ (1,1,1,9), ii) for n = 5, provided there is an integer matrix P satisfying PPT = diag (S1,S2,S3,S4), iii) for n = 7, with the extra condition that (S1,S2,S3,S4) ≠ (1,1,1,25), and possibly (S1,S2,S3,S4) =I- (1,1,1,16), (1,1,8,18), (1,1,13,13), (1,4,4,9), (1,4,9,9), (1,4,10,10), (1,8,8,9), (1,9,9,9), (2,4,4,18), (2,8,9,9), (3,4,6,8), (3,6,8,9); (4,4,4,9), (4,4,9,9), (4,5,5,9), (5,5,9,9). We show the necessary conditions for the existence of an OD(4n; S1,S2) constructed using four circulant matrices are sufficient conditions for the existence of 4 - NPAF(S1,S2) sequences for all lengths ≥ n, where n = 3 or 5

On circulant and two-circulant weighing matrices

We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we construct W(148, 144), W(152, 144), W(156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer “YES”, by constructing a circulant weighing matrix of order 142 with weight 100