New maximal two-distance sets

Abstract

A two-distance set in double-struck E signd is a point set X in the d-dimensional Euclidean space such that the distances between distinct points in X assume only two different non-zero values. Based on results from classical distance geometry, we develop an algorithm to classify, for a given d, all maximal (largest possible) two-distance sets in double-struck E signd. Using this algorithm we have completed the full classification for all d≤7, and we have found one set in double-struck E sign8 whose maximality follows from Blokhuis' upper bound on sizes of s-distance sets. While in the dimensions d≤6 our classifications confirm the maximality of previously known sets, the results in double-struck E sign7 and double-struck E sign8 are new. Their counterpart in dimension d≥10 is a set of unit vectors with only two values of inner products in the Lorentz space ℝd, 1. The maximality of this set again follows from a bound due to Blokhuis. © 1997 Academic Press.