Almost Difference Sets in Nonabelian Groups

2017/09/22 by Michel, Jerod, Wang, Qi
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.07586

Abstract

We give two new constructions of almost difference sets. The first is a generic construction of (q²(q+1),q(q²-1),q(q²-q-1),q²-1) almost difference sets in certain groups of order q²(q+1) (q is an odd prime power) having (\mathbbF_q,+) as a subgroup. The construction occurs in any group of order p²(p+1) (p is an odd prime) having (\mathbbF_p²,+) 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≡ 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. \keywordsDifference set \and Almost difference set \and Nonabelian group

Related

• Nonabelian partial difference sets constructed using abelian techniques
• A Survey on Almost Difference Sets
• Semi-regular Relative Difference Sets with Large Forbidden Subgroups
• Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets
• Restrictions on parameters of partial difference sets in nonabelian\n groups