New necessary conditions on Paley‐type partial difference sets in abelian groups

Abstract

AbstractLet be an abelian group of order , where are distinct odd prime numbers. In this paper, we prove that if contains a regular Paley‐type partial difference set (PDS), then for any is congruent to 3 modulo 4 whenever is odd. These new necessary conditions further limit the specific order of an abelian group in which there can exist a Paley‐type PDS. Our result is similar to a result on abelian Hadamard (Menon) difference sets proved by Ray‐Chaudhuri and Xiang in 1997.