Almost Difference Sets in Nonabelian Groups

TL;DR

This paper introduces two new methods for constructing almost difference sets in nonabelian groups, resulting in infinite families with applications to Ramanujan graphs.

Contribution

It presents novel generic and specific constructions of almost difference sets in nonabelian groups, expanding the known classes and their applications.

Findings

Constructed infinite families of almost difference sets in nonabelian groups.

Some resulting Cayley graphs are Ramanujan graphs.

New constructions applicable to groups with additive subgroups of finite fields.

Abstract

We give two new constructions of almost difference sets. The first is a generic construction of $(q^{2}(q+1),q(q^{2}-1),q(q^{2}-q-1),q^{2}-1)$ almost difference sets in certain groups of order $q^{2}(q+1)$ ($q$ is an odd prime power) having ($\mathbb{F}_{q},+)$ as a subgroup. The construction occurs in any group of order $p^{2}(p+1)$ ($p$ is an odd prime) having ($\mathbb{F}_{p^{2}},+)$ as an additive subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields $(4p,2p+1,p,p-1)$ almost difference sets in dihedral groups of order $4p$ where $p\equiv 3 \ ({\rm mod} \ 4)$ is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained have Cayley graphs which are Ramanujan graphs. \keywords{Difference set \and Almost difference set \and Nonabelian group}