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

Finding the maximum cardinality of a 2-distance set in Euclidean space is a classical problem in geometry. Lisoněk in 1997 constructed a maximum 2-distance set in R8 with 45 points. That 2-distance set constructed by Lisoněk has a distinguished structure of a coherent configuration of type (2,2;3) and is embedded in two concentric spheres in R8. 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 Rn, without assuming any restriction on the size of the set. We prove that there exists no such example other than that of Lisoněk. 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 derive 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.