A generalization of circulant Hadamard and conference matrices

Abstract

We study the existence and construction of circulant matrices $C$ of order $n\geq2$ with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and mutually orthogonal rows. These matrices generalize circulant conference ($d=0$) and circulant Hadamard ($d=1$) matrices. We demonstrate that matrices $C$ exist for every order $n$ and for $d$ chosen such that $n=2d+2$, and we find all solutions $C$ with this property. Furthermore, we prove that if $C$ is symmetric, or $n-1$ is prime, or $d$ is not an odd integer, then necessarily $n=2d+2$. Finally, we conjecture that the relation $n=2d+2$ holds for every matrix $C$, which generalizes the circulant Hadamard conjecture. We support the proposed conjecture by computing all the existing solutions up to $n=50$.