New Maximal Two-Distance Sets

Abstract

A two-distance set in E d is a point set X in the d-dimensional Euclidean space such that the distances between distinct points in X assume only two different nonzero 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 E d . Using this algorithm we have completed the full classification for all d 7, and we have found one set in E 8 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 E 7 and E 8 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 R d, 1 . The maximality of this set again follows from a bound due to Blokhuis. Academic Press

Spherical two-distance sets

“…Lisoněk [10] confirmed the maximality and uniqueness of previously known sets for n = 4, 5, 6. For all other n the problem of uniqueness of maximal two-distance sets is open.…”

An Upper Bound for the Cardinality of an s -Distance Set in Euclidean Space

“…If p = 1, then As for the subsets in R m there is an example of a 2-distance set in R 8 whose cardinality is 8+2 2 . This example was found by Lisoněk [7] and it is on 2 concentric spheres. However it is not a tight 4-design as a Euclidean design even though its cardinality coincides with the upper bound.…”

A geometrical characterization of strongly regular graphs

“…Example 4.12. Let G be the graph obtained from Lisoněk's 45-point two-distance set in R 8 [23]. Then d = 8 and d = 36.…”

“…Example 4.13. Let X 1 , X 2 , and X 3 be three 9-point maximal 2-distance sets whose distance ratios are the golden ratio in R 4 [23]. Let G i be the graph obtained from X i .…”

“…Indeed we have a natural upper bound for the size of an s-distance set in R d , that is, |X| d+s s [2,4]. The largest cardinality of s-distance sets is known for several dimensions and s (see [7,9,12,20,23,25,26,31]). From the view point of distance sets, we want to obtain a small-dimensional representation of a graph.…”