Some non-existence and asymptotic existence results for weighing matrices

TL;DR

This paper investigates the existence and non-existence of weighing matrices with specific properties, providing new theoretical results and asymptotic existence conditions relevant to their applications.

Contribution

It establishes new non-existence results for skew-symmetric weighing matrices and improves asymptotic existence results for symmetric weighing matrices.

Findings

No skew-symmetric weighing matrices exist for certain weights and orders.

Existence of symmetric weighing matrices for large enough orders and given weights.

Improved asymptotic existence bounds for weighing matrices.

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 $k$ cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order $4n$ and weight $k$, where $n$ is an odd positive integer. Then we show that for any square $k$, there is an integer $N(k)$ such that for each $n\ge N(k)$, there is a symmetric weighing matrix of order $n$ and weight $k$. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita and Seberry.