More on the optimal arrangement of $2d$ lines in $\mathbb{C}^d$

TL;DR

This paper introduces a new family of equiangular tight frames in complex space, conjectures their universal existence, and provides computational evidence supporting this for dimensions up to 1500.

Contribution

It presents a novel infinite family of equiangular tight frames with circulant blocks and supports the conjecture of their existence for all dimensions.

Findings

Conjecture holds for d ≤ 165 via computer-assisted proof

Numerical constructions support the conjecture for d ≤ 1500

Many matrices in the family are composed of circulant blocks

Abstract

We introduce a new infinite family of $d\times 2d$ equiangular tight frames. Many matrices in this family consist of two $d\times d$ circulant blocks. We conjecture that such equiangular tight frames exist for every $d$. We show that our conjecture holds for $d\leq 165$ by a computer-assisted application of a Newton-Kantorovich theorem. In addition, we supply numerical constructions that corroborate our conjecture for $d\leq 1500$.