Generalizations of Partial Difference Sets from Cyclotomy to Nonelementary Abelian $p$-Groups

Abstract

A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory of cyclotomy. We use the partial difference sets in elementary abelian groups to generate analogous partial difference sets in nonelementary abelian groups of the form $(Z_p)^{4s} \times (Z_{p^s})^4$. It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number. We also give a generalization of subsets of Type Q, partial difference sets consisting of one fourth of the nonidentity elements from the group, to nonelementary abelian groups. Finally, we give a similar product construction of negative Latin square type partial difference sets in the additive groups of $(F_q)^{4t+2}$ for an integer $t \geq 1$. This construction results in some new parameters of strongly regular graphs.