Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let G = C n 1 × ⋯ × C n m be an abelian group of order n = n 1 ⋯ n m , where each C n t is cyclic of order n t . We present a correspondence between the (4 n , 2, 4 n , 2 n )-relative difference sets in G × Q 8 relative to the centre Z ( Q 8 ) and the perfect arrays of size n 1 × ⋯ × n m over the quaternionic alphabet Q 8 ∪ q Q 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 × Q 8 , as well as new families of Williamson and Ito Hadamard matrices with G -invariant components.