Nonexistence of tight spherical design of harmonic index 4

TL;DR

This paper establishes a new upper bound on the size of equiangular line sets in real space using semi-definite programming, leading to the proof that tight spherical designs of harmonic index 4 do not exist for dimensions three and higher.

Contribution

It introduces a novel upper bound for equiangular lines and proves the nonexistence of certain tight spherical designs of harmonic index 4 in higher dimensions.

Findings

New upper bound for equiangular lines in $\\mathbb{R}^n$

Nonexistence of tight spherical designs of harmonic index 4 for $n \\geq 3$

Application of semi-definite programming techniques

Abstract

We give a new upper bound of the cardinality of a set of equiangular lines in $\R^n$ with a fixed angle $\theta$ for each $(n,\theta)$ satisfying certain conditions. Our techniques are based on semi-definite programming methods for spherical codes introduced by Bachoc--Vallentin [J.Amer.Math.Soc.2008]. As a corollary to our bound, we show the nonexistence of spherical tight designs of harmonic index 4 on $S^{n-1}$ with $n \geq 3$.