# Maximal $2$-distance sets containing the regular simplex

## TL;DR

This paper characterizes and constructs maximal 2-distance sets in Euclidean space that include a regular simplex, revealing infinite families with specific size and dimension properties linked to prime powers.

## Contribution

It provides a necessary and sufficient condition for extending regular simplices to 2-distance sets and constructs several maximal examples, including non-spherical sets with complex combinatorial structures.

## Key findings

- Constructed infinite families of maximal 2-distance sets containing regular simplices.
- Established a size formula for maximal 2-distance sets based on prime powers.
- Identified the dimension of these sets as a function of the parameter s.

## Abstract

A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while maintaining the $m$-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only $2$ distances. We construct several $d$-dimensional maximal $2$-distance sets that contain a $d$-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical $2$-distance sets that contain both the regular simplex and the representation of a strongly resolvable design. The maximal $2$-distance set has size $2s^2(s+1)$, and the dimension is $d=(s-1)(s+1)^2-1$, where $s$ is a prime power.