Paley type partial difference sets in abelian groups

Abstract

AbstractPartial difference sets with parameters are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then or , an odd integer. In 2010, Polhill constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using nonzero squares of a finite field, we completely answer the following question: “For which odd positive integers , can we find a Paley type partial difference set in an abelian group of order ?”