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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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