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.
Categories: math.MG math.CO