Coherent configurations and triply regular association schemes obtained from spherical designs

TL;DR

This paper generalizes previous results by showing that unions of certain spherical designs form coherent configurations and establishes triple regularity for specific tight spherical designs and related structures.

Contribution

It extends the theory of association schemes to unions of spherical designs, proving they form coherent configurations and demonstrating triple regularity for specific classes.

Findings

Union of spherical designs with certain properties form coherent configurations

Proves triple regularity for tight spherical 4, 5, 7-designs

Establishes connections with mutually unbiased bases and linked symmetric designs

Abstract

Delsarte-Goethals-Seidel showed that if $X$ is a spherical $t$-design with degree $s$ satisfying $t\geq 2s-2$, $X$ carries the structure of an association scheme. Also Bannai-Bannai showed that the same conclusion holds if $X$ is an antipodal spherical $t$-design with degree $s$ satisfying $t=2s-3$. As a generalization of these results, we prove that a union of spherical designs with a certain property carries the structure of a coherent configuration. We derive triple regularity of tight spherical $4,5,7$-designs, mutually unbiased bases, linked symmetric designs with certain parameters.