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Coherent configurations and triply regular association schemes obtained from spherical designs. (English) Zbl 1206.05105
Summary: Delsarte, Goethals and Seidel showed that if $$X$$ is a spherical $$t$$-design with degree $$s$$ satisfying $$t \geqslant 2s - 2, X$$ 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.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05B30 Other designs, configurations
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##### References:
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