Physics & Computer Science
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.
Kotsireas, Ilias S. and Koukouvinos, Christos, "Orthogonal Designs of Order 32 and 64 via Computational Algebra" (2007). Physics and Computer Science Faculty Publications. 77.