Maximal m-distance sets containing the representation of the Hamming graph H(n,m)

A set X in the Euclidean space Rd is an m-distance set if the set of Euclidean distances between two distinct points in X has size m. An m-distance set X in Rd is maximal if there does not exist a vector x in Rd such that the union of X and {x} still has only m distances. Bannai et al. (2012) investigated maximal m-distance sets that contain the Euclidean representation of the Johnson graph J(n,m). In this paper, we consider the same problem for the Hamming graph H(n,m). The Euclidean representation of H(n,m) is an m-distance set in Rm(n−1). We prove that if the representation of H(n,m) is not maximal as an m-distance set for some m, then the maximum value of n is m2+m−1. Moreover we classify the largest m-distance sets that contain the representation of H(n,m) for n≥2 and m≤4. We also classify the maximal 2-distance sets that are in R2n−1 and contain the representation of H(n,2) for n≥2.