On the isomorphism of certain primitive Q-polynomial not P-polynomial association schemes

Abstract

In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H(3,q2), q even, with respect to a symplectic polar space W(3,q) embedded in it.In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of PGL(2,q2), q even, on a non-degenerate conic of PG(2,q2) extended in PG(2,q4). Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs.