Generalised Hadamard matrices which are developed modulo a group

AbstractLet N and G be finite groups with orders n and g, respectively, and let q be a prime power. Also, let EA(q) be the elementary abelian group of order q, and let EA(n) be the group of order n which is the direct product of elementary abelian groups. This paper discusses generalised Hadamard matrices, GH(n; G), which are developed modulo a group N. These matrices have been called N-invariant GH-matrices and they are equivalent to G-relative difference sets, RDS(g, n, n, 0, n/g), modulo the direct sum of N and G. Contained in this paper are simple constructions for GH(q; EA(q)), q odd, developed modulo EA(q), and GH(q2; G), developed modulo EA(q2). Also, an algebraic setting for the study of these designs is developed, and non-existence results are obtained. Indeed, in all but 15 of the 108 cases with n⩽50, the existence of a GH(n; EA(q)), developed modulo EA(n), is either proved or disproved. In addition, a result on the non-existence of generalised weighing matrices developed modulo a group is presented. Finally a connection with error-correcting codes is presented, and it is proved that if n⩾g(g−1) then every element of G must appear in any GH(n; G) developed modulo a group