Euclidean designs obtained from spherical embedding of coherent configurations

Abstract

AbstractCoherent 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 ‐design (on two concentric spheres) in terms of the Krein numbers for . 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.