Proper circulant weighing matrices of weight $p^2$

Abstract

A circulant weighing matrix $CW(v,n)$ is a circulant matrix $M$ of order $v$ with $0,\pm 1$ entries such that $MM^T=nI_v$. In this paper, we study proper circulant matrices with $n=p^2$ where $p$ is an odd prime divisor of $v$. For $p\ge 5$, it turns out that to search for such circulant matrices leads us to two group ring equations and by studying these two equations, we manage to prove that no proper $CW(pw,p^2)$ exists when $p\equiv 3\pmod {4}$ or $p=5$.