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