A simple construction of complex equiangular lines

TL;DR

This paper introduces a straightforward method to construct maximum-sized sets of equiangular lines in complex spaces, revealing a new link with Hadamard matrices and successfully applying it to dimensions 2, 3, and 8.

Contribution

It presents a novel, simple construction for maximum equiangular lines in complex spaces, connecting with Hadamard matrices and achieving results in specific dimensions.

Findings

Constructs maximum-sized equiangular lines in dimensions 2, 3, and 8.

Establishes a new connection between equiangular lines and Hadamard matrices.

Provides a simple, explicit construction method.

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 $d \geq 2$. 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.