Weighing Matrices from Generalized Hadamard Matrices by $2$-Adjugation

Abstract

In 1988, Sarvate and Seberry introduced a new method of construction for the family of weighing matrices $W(n^2(n-1),n^2)$, where $n$ is a prime power. We generalize this result, replacing the condition on $n$ with the weaker assumption that a generalized Hadamard matrix $GH(n;G)$ exists with $|G|=n$, and give conditions under which an analogous construction works for $|G|<n$. We generalize a related construction for a $W(13,9)$, also given by Sarvate and Seberry, producing a whole new class. We build further on these ideas to construct several other classes of weighing matrices.