Four-point semidefinite bound for equiangular lines

TL;DR

This paper introduces a new four-point semidefinite bound for equiangular lines in Euclidean spaces, improving upper bounds in many dimensions and establishing uniqueness results for certain cases.

Contribution

It develops a simplified four-point semidefinite constraint method and applies it to derive tighter bounds and uniqueness results for equiangular lines.

Findings

Improved upper bounds for equiangular lines in many dimensions.

Proved uniqueness of maximum configurations for specific dimensions and angles.

Introduced a simpler alternative to existing semidefinite constraints.

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 $7 \leq d \leq 14$ with inner product $\alpha = 1/3$, and for $23 \leq d \leq 64$ with $\alpha = 1/5$.