Constructions of maximum few-distance sets in Euclidean spaces

TL;DR

This paper classifies the largest few-distance sets in low-dimensional Euclidean spaces using graph generation and algebraic methods, and constructs new examples for higher dimensions.

Contribution

It introduces a combined computational approach to classify maximum s-distance sets and provides new constructions and verifications in Euclidean spaces.

Findings

Classified largest 3-distance sets in R^4, 4-distance sets in R^3, and 6-distance sets in R^2.

Constructed new large s-distance sets for dimensions up to 8 and s up to 6.

Verified several previously known results independently.

Abstract

A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gr\"obner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.