Euclidean designs obtained from spherical embedding of coherent configurations

Abstract

Coherent configurations are a generalization of association schemes. Motivated by the recent study of Q-polynomial coherent configurations, in this paper, we study the spherical embedding of a Q-polynomial coherent configuration into some eigenspace by a primitive idempotent. We present a necessary and sufficient condition when the embedding becomes a Euclidean $t$-design (on two concentric spheres) in terms of the Krein numbers for $t\le 4$. In addition, we obtain some Euclidean 2- or 3-designs from spherical embedding of coherent configurations including tight relative 4- or 5-designs in binary Hamming schemes and the union of derived designs of a tight 4-design in Hamming schemes.