Equiangular lines and spherical codes in Euclidean space

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 bound is achieved if and only if $\theta = \arccos \frac{1}{3}$. Indeed, we show that for all $\theta \ne \arccos {\frac{1}{3}}$ and and sufficiently large n, the number of equiangular lines is at most 1.93n. 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.