Proper circulant weighing matrices of weight \(p^2\)
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 \(p=5\).
weighing matrices circulant matrices group rings 05b10 05b15