| Turyn's classical paper in 1965 established the link between character theory and difference sets. Since this time, people have tried showing existence or nonexistence of difference sets in certain groups. Due to their link to Hadamard matrices, the existence of Hadamard difference sets has been of particular interest. In 1991, Davis showed existence of Hadamard difference sets in the group C2r+2 x C2r. Using the techniques developed by Davis and using Turyn's work on nonexistence, Kraemer completely resolved existence of difference sets in abelian 2-groups. While much work has been done on determining the existence of Hadamard difference sets, many questions remain open. For instance, the existence of reversible difference sets is open for many groups. Also, recent work has been done on DRAD difference sets. Many questions remain open for DRAD difference sets.;In an effort to solve the existence question in certain groups, we develop a theory for constructing difference sets. The theory uses characters, aspects of Fourier analysis, Galois theory, and algebraic number theory. We use these tools to create tiling structures that produce Hadamard difference sets in the group C2r+2 x C2r. Tiling structures that produce reversible and DRAD difference sets the group C2 r x C2r are also created. Existence of difference sets in these groups was previously known, but shown by different methods. Using the same theory, we establish nonexistence results on DRAD difference sets that are not previously known, and also establish nonexistence of difference sets in a group of order 576. In the last chapter, we give two spread constructions, which explain all difference sets in groups of order 36. |
