New constructions of optimal arrangements of $2d$ lines in ${\mathbb{C}}^d$

Abstract

In this paper we provide new constructions of equiangular tight frames of size $2d$ in ${\mathbb{C}}^d$. We generalize the doubling construction of Fallon and Iverson to a tensor multiplication construction based on a suitable pair consisting of a complex Hadamard matrix and an equiangular tight frame. In particular, such a pair always exists whenever there is an amicable pair of real Hadamard matrices. Most notably, amicable Hadamard pairs of order $q+1$ exist for all prime powers $q\equiv 3(\mathrm{mod}4)$. We also find specific constructions based on a family of pairs of order 6 and on pairs whose equiangular tight frames are defined by Paley conference matrices with $q\equiv 1(\mathrm{mod}4)$. Finally, we provide a power construction of equiangular tight frames that generalizes the construction of Turyn for conference matrices.