Possible indices for a class of association schemes with thin residue equal to the thin radical

Abstract

We consider association schemes with thin radical isomorphic to an elementary abelian p-group of rank 2, such that the thin residue coincides with the thin radical and all non-thin elements have valency p. We show that when the order of the thin quotient of such schemes exceeds p2, it must be equal to p2+p+1 (a known upper bound), and we show that there exist examples whose thin quotient has order p2−1. We also show that if the thin quotient has order at least 10, then one can always find a non-Schurian example with the same thin quotient.