Constructions of maximum few-distance sets in Euclidean spaces

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 Gröbner 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 $d\le 8$ and $s\le 6$, and independently verify several earlier results from the literature.

9 pages, preprint