Weighing Matrices from Generalized Hadamard Matrices by \(2\)-Adjugation

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.