Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$

It is known that a real symmetric circulant matrix with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d\geq0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with $d$ different from an odd integer is $n=2d+2$. We also discuss a similar problem for symmetric circulant matrices defined over finite rings $\mathbb{Z}_m$. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.