Partial Difference Sets with Paley Parameters

Abstract

Partial difference sets with parameters (ν,k,λ,μ) = (ν,(ν − 1)/2, (ν − 5)/4,(ν − 1)/4) are called Paley partial difference sets. By using finite local rings, we construct a family of Paley PDSs for abelian p-groups with any given exponent. Furthermore, we prove some non-existence results on Paley PDSs. Using these results, we prove that Paley PDSs exist in a rank 2 abelian group if and only if the group is isomorphic to ℤpr×ℤpr where p is an odd prime.