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.