A simple construction of complex equiangular lines

Abstract

A set of vectors of equal norm in ${\mathbb{C}}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in ${\mathbb{C}}^d$ for every $d\ge 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.

11 pages; minor revisions and comments added in section 1 describing a link to previously known results; correction to Theorem 1 and updates to references