Disjoint and external partial difference families and cyclotomy

Abstract

In this Thesis, we introduce two new combinatorial objects known as Disjoint Partial Difference Families and External Partial Difference Families: these objects generalise Disjoint Difference Families (DDFs), External Difference Families (EDFs) and Partial Difference Sets (PDSs), which have all been well-studied in the literature. We demonstrate how DPDFs and EPDFs can be formed from PDSs and Relative Difference Sets (RDSs), presenting both cyclotomic and non-cyclotomic constructions of these objects.

We also develop two new cyclotomic frameworks within this Thesis, which allow us to identify new cyclotomic constructions of DPDFs and EPDFs along with other types of difference structures. The first of these cyclotomic frameworks relies upon a series of partition results, the second utilises natural connections between cyclotomic numbers and cyclotomic cosets. These frameworks remove the need to evaluate all cyclotomic numbers in a particular finite field.

We primarily use these frameworks to identify new DPDF and EPDF constructions, however, we also use the cyclotomic techniques underpinning these frameworks to establish a series of algorithms that compute the cyclotomic numbers in a given finite field. Further, we use one of these frameworks to prove that a PDS with Denniston parameters exists in the group Z_3^9: as 3 is an odd prime, it was previously believed that such a PDS would not exist in this group.