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

Abstract

A set $X$ in the Euclidean space ${\mathbb{R}}^d$ is called 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 ${\mathbb{R}}^d$ is said to be maximal if there does not exist a vector $x$ in ${\mathbb{R}}^d$ such that the union of $X$ and $\{x\}$ still has only $m$ distances. Bannai--Sato--Shigezumi (2012) investigated the maximal $m$-distance sets which 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 ${\mathbb{R}}^{m(n-1)}$. We prove that the maximum $n$ is $m^2+m-1$ such that the representation of $H(n,m)$ is not maximal as an $m$-distance set. Moreover we classify the largest $m$-distance sets which contain the representation of $H(n,m)$ for $m\le 4$ and any $n$. We also classify the maximal $2$-distance sets in ${\mathbb{R}}^{2n-1}$ which contain the representation of $H(n,2)$ for any $n$.

19 pages, no figure

References in corpus (2)

• Uniqueness of maximum three-distance sets in the three-dimensional Euclidean space
• Maximal $m$-distance sets containing the representation of the Johnson graph $J(n,m)$