Paley type group schemes from cyclotomic classes and Arasu-Dillon-Player difference sets

Abstract

In this paper, we present constructions of abelian Paley type group schemes by using multiplicative characters of finite fields and Arasu-Dillon-Player difference sets. The constructions produce many new Paley type group schemes that were previous unknown in our classification of Paley type group schemes in finite fields of small orders.

Three-class association schemes from cyclotomy

“…Quite recently, there have been several constructions of strongly regular graphs with new parameters and skew Hadamard difference sets from cyclotomy, the latter giving rise to two-class nonsymmetric association schemes, see [16,18,20,23] for strongly regular graphs and [9,17,18,24] for skew Hadamard difference sets. In [27], the authors discussed the problem when a Cayley graph on a finite field with a single cyclotomic class as its connection set can form a strongly regular graph.…”

“…Furthermore, Muzychuk [26] constructed infinitely many inequivalent skew Hadamard difference sets in elementary abelian groups of order q 3 . Recently, in [9,17,18,24], the authors constructed further counterexamples of this conjecture by taking suitably a union of cyclotomic classes. See the introduction of [17] (or [9]) for a short survey on skew Hadamard difference sets.…”

“…Recently, in [9,17,18,24], the authors constructed further counterexamples of this conjecture by taking suitably a union of cyclotomic classes. See the introduction of [17] (or [9]) for a short survey on skew Hadamard difference sets.…”