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

Abstract

We show the optimal coherence of $2d$ lines in ${\mathbb{C}}^d$ is given by the Welch bound whenever a skew Hadamard of order $d+1$ exists. Our proof uses a variant of Hadamard doubling that converts any equiangular tight frame of size $\frac{d-1}{2}\times d$ into another one of size $d\times 2d$. Among $d<150$, this produces equiangular tight frames of new sizes when $d=11$, $35$, $39$, $43$, $47$, $59$, $67$, $71$, $83$, $95$, $103$, $107$, $111$, $119$, $123$, $127$, $131$, and $143$.