Almost Difference Sets in Nonabelian Groups

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 $p\equiv 3(\mathrm{mod}4)$ 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}

References in corpus (1)

• On Some Expander Graphs and Algebraic Cayley Graphs