Bounds on three- and higher-distance sets

TL;DR

This paper improves bounds on the maximum size of s-distance sets in various metric spaces, including spheres, Hamming, and Johnson spaces, with exact results for specific cases.

Contribution

It provides improved upper bounds for 3- and 4-distance sets in n-spheres and determines exact maximum sizes for certain dimensions, advancing understanding of s-distance set limitations.

Findings

Maximum cardinalities of 3-distance sets in 7- and 21-dimensional spheres.

Enhanced upper bounds for 3- and 4-distance sets in n-spheres.

Maximum sizes of s-distance sets in Hamming and Johnson spaces for specific s and dimensions.

Abstract

A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for fixed s and M. In this paper, we improve the known upper bound for s-distance sets in n-sphere for s=3,4. In particular, we determine the maximum cardinalities of three-distance sets for n=7 and 21. We also give the maximum cardinalities of s-distance sets in the Hamming space and the Johnson space for several s and dimensions.