On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)

TL;DR

This paper proves that the only 2-distance set in Euclidean space with a coherent configuration of type (2,2;3) is the one constructed by Lisoněk in 1997, using geometric and Diophantine analysis.

Contribution

It establishes the uniqueness of Lisoněk's 2-distance set in Euclidean space among all similar coherent configurations without size restrictions.

Findings

Lisoněk's 2-distance set in R^8 is unique for coherent configurations of type (2,2;3).

The proof involves solving Diophantine equations derived from geometric embeddings.

No other embeddings of this type exist beyond Lisoněk's example.

Abstract

Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lison\v{e}k in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by Lison\v{e}k has a distinguished structure of a coherent configuration of type $(2,2;3)$ and is embedded in two concentric spheres in $\mathbb R^8$. In this paper we study whether there exists any other similar embedding of a coherent configuration of type $(2,2;3)$ as a $2$-distance set in $\mathbb R^n$, without assuming any restriction on the size of the set. We prove that there exists no such example other than that of Lison\v{e}k. The key ideas of our proof are as follows: (i) study the geometry of the embedding of the coherent configuration in Euclidean spaces and to drive diophantine equations coming from this embedding. (ii) solve diophantine equations with certain additional conditions of integrality of some parameters of the combinatorial structure by using the method of auxiliary equations.