Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Abstract

Let $G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}$ be an abelian group of order $n=n_1\dots n_m$, where each $\mathbf{C}_{n_t}$ is cyclic of order $n_t$. We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in $G\times Q_8$ relative to the centre $Z(Q_8)$ and the perfect arrays of size $n_1\times \dots \times n_m$ over the quaternionic alphabet $Q_8\cup qQ_8$, where $q=(1+i+j+k)/2$. In view of this connection, for $m=2$ we introduce new families of relative difference sets in $G\times Q_8$, as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.