Pseudocyclic association schemes arising from the actions of PGL(2,2^m) and P\Gamma L(2,2^m)

Abstract

The action of $PGL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ affords an association scheme, which was shown to be pseudocyclic in Hollmann's thesis in 1982. It was further conjectured in Hollmann's thesis that the orbital scheme of $P\Gamma L(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ is also pseudocyclic if $m$ is an odd prime. We confirm this conjecture in this paper. As a by-product, we obtain a class of Latin square type strongly regular graphs on nonprime-power number of points.