A Family of Partial Difference Sets with Denniston Parameters in Nonelementary Abelian 2-Groups

A k-element subset D of a finite multiplicative group G of order v is called a (v, k, λ, μ)partial difference set in G (PDS) provided that the multiset of ‘differences’ {d1d −1 2 | d1, d2 ∈ D, d1 6= d2} contains each nonidentity element of D exactly λ times and each nonidentity element in G\D exactly μ times. See [11] for background on partial difference sets. We will limit our attention to abelian groups in this paper. In that context, a character of an abelian group is a homomorphism from the group to the multiplicative group of complex roots of unity. The principal character is the character mapping every element of the group to 1. All other characters are called nonprincipal. Starting with the important work of Turyn [16], character sums have been a powerful tool in the study of difference sets of all types. The following lemma states how character sums can be used to verify that a subset of a group is a PDS.