On Symmetric Weighing Matrices

Abstract

Symmetric weighing matrices are an important class of combinatorial designs and their constructions remains unknown even for some small orders. The existence of symmetric weighing matrices becomes an even more interesting challenge as these matrices possess both beautiful mathematical properties and many applications. In this paper, we present two original construction methods for symmetric weighing matrices. The suggested methods lead to two infinite families of symmetric weighing matrices. The first consists of symmetric weighing matrices W(2p×15,2q×25) for all q<p and p=1,2,… while the second is an infinite family of symmetric weighing matrices W(2p+1×15,2+2q×25) for all q≤p and p=1,2,…. These matrices are constructed by combining together a circulant and a negacyclic matrix with identical first row. The first of the infinite families includes a symmetric weighing matrix of order 30 and weight 25. The results presented here are new and have never been reported.