#### Abstract

Higher-dimensional orthogonal designs of type (l,l)n are used to obtain higher-dimensional weighing matrices of type (q)n, side q+l and propriety (2,2,...,2) for q = l(mod 4) a prime power. Next, n-dimensional orthogonal designs of type (l,l,l,l)n, side 4 and propriety (2,2, ...,2) are constructed. These are then used to show that higher-dimensional Hadamard matrices of order (4t)t exist whenever t is the side of 4-Williamson matrices. This establishes the existence of higher-dimensional Hadamard matrices of order (4t) t for t odd, 1 ≤ t ≤ 33 and several infinite families, all of propriety (2,2,...,2). Finally, we establish that if there is an Hadamard matrix that can be obtained from a group difference set with parameters (4s 2, 2s2±s, s2±s) then there is a higher-dimensional Hadamard matrix of order (4s2)n.

## Publication Details

Hammer, J, and Seberry, J, Higher-dimensional orthogonal designs and Hadamard matrices II, Proceedings of the Nineth Manitoba Conference on Numerical Mathematics, Congressus Numerantium, 27, 1979, 23-29.