Paley type partial difference sets in abelian groups

TL;DR

This paper characterizes the possible orders of abelian groups containing Paley type partial difference sets, showing they must be of the form n^4 or 9n^4, and confirms constructions exist for these orders.

Contribution

It proves that Paley type partial difference sets in abelian groups only exist in groups of order n^4 or 9n^4, completing the classification for odd positive integers.

Findings

Existence of Paley type partial difference sets only in groups of order n^4 or 9n^4.

Confirmed constructions for these orders by Polhill and classical Paley methods.

Complete classification for odd v > 1 regarding such difference sets.

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 $v$?"