Coherent configurations and triply regular association schemes obtained from spherical designs

Delsarte, Goethals and Seidel showed that if X is a spherical t-design with degree s satisfying t ⩾ 2 s − 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 = 2 s − 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.