New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-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. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G? when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form $${G = (Z_2)^{4s_0} \times (Z_4)^{2s_1} \times (Z_{16})^{4s_2} \times \cdots \times (Z_{2^{2r}})^{4s_r}}$$ where the s i are nonnegative integers and s 0 + s 1 ? 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form $${G = (Z_3)^2 \times (Z_3)^{2{s_1}} \times (Z_9)^{2{s_2}} \times \cdots \times (Z_{3^{2k}})^{2{s_k}}}$$ for nonnegative integers s i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.