Disjoint weighing matrices (Q2114772)

A weighing matrix of order \(n\) and weight \(w\), denoted by \(W(n,w)\), is an \(n \times n\) \((0, \pm 1)\)-matrix \(W\) such that \(WW^{\top}=wI_n\), where \(I_n\) denotes the identity matrix of order \(n\). In this paper, the authors introduce the notion of disjoint weighing matrices, as a generalization of orthogonal designs. Disjoint weighing matrices can be used in the construction of Bush-type Hadamard matrices, orthogonal designs, symmetric group divisible designs, and disjoint weighing designs. The main focus of the paper is on skew-symmetric disjoint weighing matrices. Using a recursive construction and a computer search, the authors obtain infinite classes of disjoint weighing matrices. They also introduce a generalization of symmetric group divisible designs and use skew-symmetric disjoint weighing matrices to obtain commutative association schemes with 3 or 4 classes.