Constructions of maximum few-distance sets in Euclidean spaces

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