Almost difference sets in nonabelian groups

Jerod Michel, Qi Wang

Abstract: We give two new constructions of almost difference sets. The first is a generic construction of $$(q^{2}(q+1),\,q(q^{2}-1),\,q(q^{2}-q-1),\,q^{2}-1)$$ ( 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 2 ( q + 1 ) (q is an odd prime power) having ( $$\mathbb {F}_{q^{2}},\,+)$$ F q 2 , + ) as a subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields $$(4p,\,2p+1,\,p,\,p-1)$$ ( 4 p , 2 p + 1 , p , p - 1 ) almost difference sets in dihedral groups of order 4p where $$p\equiv 3 \ (\mathrm{mod} \ 4)$$ p ≡ 3 ( mod 4 ) is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained produce Cayley graphs which are Ramanujan graphs.