| Abstract | We detail the Williamson array construction based on quaternions, following the description by Baumert and Hall. By analogy, we extend the construction to larger arrays using matrix representations of the algebras of octonions and sedenions. In the case of octonions, we obtain the full orthogonal design OD(8;1,1,1,1,1,1,1,1) or order 8 with 8 variables. In the case of sedenions we obtain the full orthogonal design OD(16;1,1,7,7) of order 16 with 4 variables and the full orthogonal design OD(16;1,1,2,2,2,2,2,2,2) of order 16 with 9 variables. We use OD(16;1,1,2,2,2,2,2,2,2) to search for inequivalent Hadamard matrices of orders 112, 144, 176 and we establish constructively three new lower bounds for the numbers of inequivalent Hadamard matrices of these three orders. |
