Paley type partial difference sets in abelian groups

Abstract

Partial difference sets with parameters $(v,k,\lambda ,\mu )=(v,(v-1)/2,(v-5)/4,(v-1)/4)$ 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 $G$ of an order not a prime power, then $|G|=n^4$ or $9n^4$, where $n>1$ is an odd integer. In 2010, Polhill \cite{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 non-zero squares of a finite field, we completely answer the following question: "For which odd positive integer $v>1$, can we find a Paley type partial difference set in an abelian group of order $v$?"