Table of Contents
International Journal of Combinatorics
Volume 2016 (2016), Article ID 2162849, 6 pages
http://dx.doi.org/10.1155/2016/2162849
Research Article

Some Nonexistence and Asymptotic Existence Results for Weighing Matrices

Lassonde School of Engineering, York University, Toronto, ON, Canada M3J 1P3

Received 4 January 2016; Accepted 18 February 2016

Academic Editor: Christos Koukouvinos

Copyright © 2016 Ebrahim Ghaderpour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order and weight , where is an odd positive integer. Then we show that, for any square , there is an integer such that, for each , there is a symmetric weighing matrix of order and weight . Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.