Semidefinite Programming Bounds For Spherical Three-distance Sets

TL;DR

This paper employs semidefinite programming to improve upper bounds on the size of spherical three-distance sets in various dimensions, notably establishing the maximum size as 2300 in a023.

Contribution

It introduces new bounds for spherical three-distance sets using semidefinite programming, including the exact maximum size in a023.

Findings

Improved upper bounds in a027, a020, a021, a023, a024, a025.

Maximum size of spherical three-distance sets in a023 is 2300.

Semidefinite programming effectively tightens bounds for these geometric configurations.

Abstract

A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that for each pair of distinct vectors has three inner product values. We use the semidefinite programming method to improve the upper bounds of spherical three-distance sets for several dimensions. We obtain better bounds in $\mathbb{R}^7$, $\mathbb{R}^{20}$, $\mathbb{R}^{21}$, $\mathbb{R}^{23}$, $\mathbb{R}^{24}$ and $\mathbb{R}^{25}$. In particular, we prove that maximum size of spherical three-distance sets is $2300$ in $\mathbb R^{23}$.