Geodesic
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$
Communications in Mathematics, Tome 29 (2021) no. 1, pp. 15-34.

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It is known that a real symmetric circulant matrix with diagonal entries $d\geq 0$, off-diagonal entries $\pm 1$ 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 $d\geq 0$ 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.
Classification : 15B05, 15B10, 15B36
Keywords: Circulant matrix; orthogonal matrix; Hadamard matrix; mutually unbiased base 
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     author = {Contreras, Daniel Uzc\'ategui and Goyeneche, Dardo and Turek, Ond\v{r}ej and V\'aclav{\'\i}kov\'a, Zuzana},
     title = {Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$},
     journal = {Communications in Mathematics},
     pages = {15--34},
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Contreras, Daniel Uzcátegui; Goyeneche, Dardo; Turek, Ondřej; Václavíková, Zuzana. Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$. Communications in Mathematics, Tome 29 (2021) no. 1, pp. 15-34. http://geodesic.mathdoc.fr/item/COMIM_2021__29_1_a1/