New upper bound for the cardinalities of $s$-distance sets on the unit sphere

TL;DR

This paper introduces a new upper bound for the size of $s$-distance sets on the unit sphere, surpassing existing bounds like Fisher's inequality and applicable where linear programming bounds fail.

Contribution

The authors derive a novel upper bound for $s$-distance sets on spheres, improving upon classical bounds and extending applicability beyond linear programming limitations.

Findings

New upper bound for $s$-distance sets on spheres

Improves Fisher type inequality

Applicable to cases where linear programming bound does not apply

Abstract

We have the Fisher type inequality and the linear programming bound as upper bounds for the cardinalities of $s$-distance sets on $S^{d-1}$. In this paper, we give a new upper bound for the cardinalities of $s$-distance sets on $S^{d-1}$ for any $s$. This upper bound improves the Fisher typer inequality and is useful for $s$-distance sets which are not applicable to the linear programming bound.