Uniform bound on the strength of Q-polynomial spherical embeddings

Establish an absolute constant K>0 such that the spherical embedding with respect to a corresponding Q-polynomial idempotent of every Q-polynomial association scheme with multiplicity m≥3 has strength at most K, and determine whether K=11 is admissible.

Background

The paper proves that, for a P- and Q-polynomial association scheme with at least three classes and a corresponding Q-polynomial idempotent of multiplicity at least three, the associated spherical embedding has strength at most five. The concluding problem asks whether any Q-polynomial association scheme, without the P-polynomial hypothesis, admits a uniform upper bound on the strength of its spherical embedding.

The proposed value K=11 is motivated by the known restriction that tight spherical t-designs in dimensions m≥3 can occur only for t∈{1,2,3,4,5,7,11}. The question is presented as a dual analogue of the conjectured universal upper bound on the girth of distance-regular graphs.