New maximal two-distance sets.

Journal of Combinatorial Theory A 77 (1997), 318-338.

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 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 $E^d$. Using this algorithm we have completed the full classification for all $d\le 7$, 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 $d\le 6$ our classifications confirm the maximality of previously known sets, the results in $E^7$ and $E^8$ are new. Their counterpart in dimension $d\ge 10$ 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.