Generalised Hadamard matrices which are developed modulo a group

Abstract

Let 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( q 2; G), developed modulo EA( q 2). 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.