Equiangular Lines and Spherical Codes in Euclidean Space

TL;DR

This paper establishes upper bounds on the number of equiangular lines in high-dimensional Euclidean spaces for fixed angles, confirming conjectures and extending results to multiple angles and spherical codes.

Contribution

It proves that for fixed angles, the maximum number of equiangular lines in  space is at most 2n-2, achievable only at a specific angle, and extends bounds to multiple angles and spherical codes.

Findings

Maximum of 2n-2 lines for fixed angle  space

Achievability only at os(1/3) angle

Bound of O(n^k) lines for k fixed angles

Abstract

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only for $\theta = \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k)$ lines in $\mathbb{R}^n$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.