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

TL;DR

This paper investigates the maximality of Euclidean representations of Hamming graphs as m-distance sets in Euclidean space, establishing bounds on n and classifying maximal sets for small m and specific cases.

Contribution

It extends the study of maximal m-distance sets to include the Euclidean representations of Hamming graphs, providing bounds and classifications for small parameters.

Findings

Maximum n for non-maximal representation is m^2 + m - 1.

Classified largest m-distance sets containing Hamming graph representations for m ≤ 4.

Classified maximal 2-distance sets containing Hamming graph representations in specific dimensions.

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 $m\leq 4$ 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 $n$.