Four-point semidefinite bound for equiangular lines

Abstract

A set of lines in ${\mathbb{R}}^d$ passing through the origin is called equiangular if any two lines in the set form the same angle. We proved an alternative version of the three-point semidefinite constraints developed by Bachoc and Vallentin, and the multi-point semidefinite constraints developed by Musin for spherical codes. The alternative semidefinite constraints are simpler when the concerned object is a spherical $s$-distance set. Using the alternative four-point semidefinite constraints, we found the four-point semidefinite bound for equiangular lines. This result improves the upper bounds for infinitely many dimensions $d$ with prescribed angles. As a corollary of the bound, we proved the uniqueness of the maximum construction of equiangular lines in ${\mathbb{R}}^d$ for $7\le d\le 14$ with inner product $\alpha =1/3$, and for $23\le d\le 64$ with $\alpha =1/5$.