On the Non-Existence of \((g, g\lambda – 1; \lambda)\)-Difference Matrices

Abstract

A \((g,k; \lambda)\)-difference matrix over the group \((G, o)\) of order \(g\) is a \(k\) by \(g\lambda\) matrix \(D = (d_{ij})\) with entries from \(G\) such that for each \(1 \leq i < j \leq k\), the multiset \(\{d_{il}\) o \(d_{jl}^{-1} \mid 1 \leq l \leq g\lambda\}\) contains every element of \(G\) exactly \(\lambda\) times. Some known results on the non-existence of generalized Hadamard matrices, i.e., \((g,g\lambda; \lambda)\)-difference matrices, are extended to \((g, g-1; \lambda)\)-difference matrices.