New bounds for equiangular lines

Abstract

A set of lines in ${\mathbb{R}}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in ${\mathbb{R}}^n$, using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtained in dimensions $24\le n\le 136$. In particular, we show that the maximum number of equiangular lines in ${\mathbb{R}}^n$ is $276$ for all $24\le n\le 41$ and is 344 for $n=43.$ This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973).

Minor corrections; added one new reference. To appear in "Discrete Geometry and Algebraic Combinatorics," A. Barg and O. R. Musin, Editors, Providence: RI, AMS (2014). AMS Contemporary Mathematics series