On the Nonexistence of Partial Difference Sets by Projections to Finite Fields (Communications in Mathematical Research, 2022)
In the study of (partial) difference sets and their generalizations ingroups $G$, the most widely used method is to translate their definition into anequation over group ring $\mathbb{Z}[G]$ and to investigate this equation by applyingcomplex representations of $G.$ In this paper, we investigate the existence of(partial) difference sets in a different way. We project the group ring equationsin $\mathbb{Z}[G]$ to $\mathbb{Z}[N]$ where $N$ is a quotient group of $G$ isomorphic to the additivegroup of a finite field, and then use polynomials over this finite field to derivesome existence conditions.
Keywords: Partial difference set; strongly regular graph; finite field