On the Nonexistence of Partial Difference Sets by Projections to Finite Fields

Abstract

In the study of (partial) difference sets and their generalizations in groups $G$, the most widely used method is to translate their definition into an equation over group ring $\mathbb{Z}[G]$ and to investigate this equation by applying complex representations of $G.$ In this paper, we investigate the existence of (partial) difference sets in a different way. We project the group ring equations in $\mathbb{Z}[G]$ to $\mathbb{Z}[N]$ where $N$ is a quotient group of $G$ isomorphic to the additive group of a finite field, and then use polynomials over this finite field to derive some existence conditions.