# New necessary conditions for Payley type PDS in Abelian groups

## TL;DR

This paper establishes new necessary conditions on the prime factorization of the order of Abelian groups that can contain Paley type partial difference sets, specifically linking prime congruences to exponents.

## Contribution

It provides novel necessary conditions on the prime factors of group order for the existence of Paley type PDS in Abelian groups, extending previous results.

## Key findings

- If a regular Paley type PDS exists, each prime with odd exponent must be congruent to 3 mod 4.
- These conditions restrict the possible orders of Abelian groups admitting such PDS.
- The results are analogous to known conditions for Abelian Hadamard difference sets.

## Abstract

In this paper we prove that if there is a regular Paley type partial difference set in an Abelian group $G$ of order $v$, where $v=p_1^{2k_1}p_2^{2k_2}\cdots p_n^{2k_n}$, $n\ge 2$, $p_1$, $p_2$, $\cdots$, $p_n$ are distinct odd prime numbers, then for any $1 \le i \le n$, $p_i$ is congruent to 3 modulo 4 whenever $k_i$ is odd. These new necessary conditions further limit the specific order of an Abelian group $G$ in which there can exist a Paley type partial difference set. Our result is similar to a result on Abelian Hadamard (Menon) difference sets proved by Ray-Chaudhuri and Xiang in 1997.