Symmetric weighing matrices constructed using group matrices

Abstract

A weighing matrix of order n and weight m 2 is a square matrix M of order n with entries from {-1,0,+1} such that MM T =m 2 I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M T =M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p r and weight p 2 where p is a prime and p ≥ 5. © 2005 Springer Science+Business Media, Inc.