^{1} Department of Mathematics, Technical University of Denmark^{2} Department of Applied Mathematics and Computer Science, Technical University of Denmark

Abstract:

A two-distance set in E^d is a point set X inthe d-dimensional Euclidean spacesuch that the distances between distinct points in Xassume only two different non-zero values. Based on results from classical distance geometry, we developan algorithm to classify, for a given dimension, all maximal (largest possible)two-distance sets in E^d.Using this algorithm we have completed the full classificationfor all dimensions less than or equal to 7, andwe have found one set in E^8 whosemaximality follows from Blokhuis' upper bound on sizes of s-distance sets.While in the dimensions less than or equal to 6 our classifications confirmthe maximality of previously known sets, the results in E^7 and E^8are new. Their counterpart in dimension larger than 10is a set of unit vectors with only two values of inner products in the Lorentz space R^{d,1}.The maximality of this set again follows from a bound due to Blokhuis.