Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames

Abstract

In this article we construct new, previously unknown parametric families of complex conference matrices of even orders and of complex Hadamard matrices of square orders and related them to complex equiangular tight frames. It is shown that for any odd integer $k\ge 3$ such that $2k=p^{\alpha }+1$, p prime, $\alpha$ non-negative integer, on the one hand there exists a (2k, k) complex equiangular tight frame and for any $\beta \in \mathbb {N}^{*}$ there exists a $((2k)^{2^{\beta }},\frac{1}{2}(2k)^{2^{\beta -1}}((2k)^{2^{\beta -1} }\pm 1))$ complex equiangular tight frame depending on one unit complex number, and on the other hand there exist a family of $((4k)^{2^{\beta }},\frac{1}{2}(4k)^{2^{\beta -1}}((4k)^{2^{\beta -1}}\pm 1))$ complex equiangular tight frames depending on two unit complex numbers.

To T. Zamfirescu on the occasion of his seventieth birthday