Difference sets in pseudocyclic association schemes

Abstract

Wilson (1972) proved that for fixed positive integers d and r such that 2d divides r(r−1), there exists a subset Y of size r of the finite field of order q such that the r(r−1) nonzero differences occurring from Y are evenly distributed to the dth cyclotomic classes provided that q is sufficiently large relative to d and r. On the other hand, Kajiura, Matsumoto and Okuda (2019) introduced the concept of difference sets in commutative association schemes in their study on a quasi-Monte Carlo method to approximate integrations over the point sets. Then, Wilson's result implies the existence of difference sets in cyclotomic schemes. In this paper, we generalize Wilson's result into pseudocyclic commutative association schemes and show the existence of difference sets in pseudocyclic association schemes with sufficiently large numbers of points.