Delsarte, Goethals and Seidel showed that if X is a spherical t-design with degree s satisfying t≥2s-2, X carries the structure of an association scheme. Also Bannai and 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 systems of symmetric designs with certain parameters. © 2010 Elsevier Inc.
Suda, S. (2010). Coherent configurations and triply regular association schemes obtained from spherical designs. Journal of Combinatorial Theory. Series A, 117(8), 1178–1194. https://doi.org/10.1016/j.jcta.2010.03.016