Constructing Hadamard Matrices with Orthogonal Pairs

A procedure based on the Kronecker product yields \(\pm 1\)-matrices \(X,Y\) of order \(8mn\), satisfying
\(XX^t + YY^t = 8mnI \quad {and} \quad XY^t = YX^t = 0,\)
given Hadamard matrices of orders \(4m\) and \(4n\). This allows the construction of some infinite classes of Hadamard matrices – and in particular orders \(8mnp\), for values of \(p\) including (for \(j \geq 0\)) \(5, 9^j, 25, 9^j, \), improving the usual Kronecker product construction by at least a factor of \(2\). A related construction gives Hadamard matrices in orders \(4 \cdot 5^i \cdot 9^j, 0 \leq i \leq 4\). To this end we introduce some disjoint weighing matrices and exploit certain Williamson matrices studied by Turyn and Xia. Some new constructions are given for symmetric and skew weighing matrices, resolving the case of skew \(W(N, 16)\) for \(N = 30, 34, 38\).