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 Gröbner basis computation to classify the largest 3-distance sets in R⁴, the largest 4-distance sets in R³, and the largest 6-distance sets in R². 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.