Subsets of groups exhibiting regularity in differences
Date
2019
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
This thesis is primarily concerned with subsets of groups that exhibit a regularity of differences (if written additively). In it, both non-existence results and existence results shall be established, along with the development of a general construction technique for generalized difference sets. Chapter 1 contains an introduction to the objects being considered as well a brief background of character theory. ☐ In Chapter 2 we prove certain integrality conditions regarding the parameters of PDS’s. This leads to a particular nonexistence result. ☐ Neo-difference sets have been used to study finite projective planes of Lenz-Barlotti type I.4. Although a nonexistence proof remains elusive, several results exist regarding conditions on orders of such projective planes. We generalize a group-ring equation used in proving one of these conditions in Chapter 3. ☐ In Chapters 4-7, we outline a method of constructing infinite families of PDS’s in finite fields and provide examples of three such constructions which come from the image sets of polynomials over said finite fields. These infinite families of PDS’s are not new, however, and Chapter 8 establishes the equivalence of these recent constructions with Maiorana- McFarland bent functions and orthogonal arrays. ☐ In Chapter 9, we provide examples of GDSs found in fields of characteristic 3 using the methods put forth in Chapter 4. Although no families of GDS’s are found, there are some possibilities worth investigating. ☐ Finally, we outline the publication status of our results. Those of Chapter 2 were subsumed by a more general result of De Winter etc., so has never been submitted to be published. The results of Chapter 3 was published in the article A Wilbrink-like equation for neo-difference sets. The results of Chapters 4 through 8 were published in the article Image Sets with Regularity of Differences. The results of Chapter 9 remain unsubmitted as they are incomplete. We hope to establish an infinite class of GDS’s that contain our work.