New bounds for spherical two-distance sets

TL;DR

This paper employs semidefinite programming to derive improved bounds on the maximum size of spherical two-distance sets in various dimensions, resolving previous divergent bounds and providing exact solutions for specific cases.

Contribution

It introduces new bounds for spherical two-distance sets using semidefinite programming, achieving exact results in certain dimensions where earlier bounds diverged.

Findings

Exact maximum sizes for n=23, 40≤n≤93 (n≠46,78)

Improved bounds over previous estimates

Resolution of divergent bounds in specific dimensions

Abstract

A spherical two-distance set is a finite collection of unit vectors in $\reals^n$ such that the set of distances between any two distinct vectors has cardinality two. We use the semidefinite programming method to compute improved estimates of the maximum size of spherical two-distance sets. Exact answers are found for dimensions $n=23$ and $40\le n\le 93\; (n\ne 46,78)$ where previous results gave divergent bounds.