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.
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机译：Turyn在1965年的经典论文中确立了角色理论与差异集之间的联系。从那时起，人们尝试在某些组中显示差异集的存在或不存在。由于它们与Hadamard矩阵的联系，因此特别引起了Hadamard差集的存在。 1991年，戴维斯（Davis）显示了C2r + 2 x C2r组中Hadamard差集的存在。使用戴维斯（Davis）开发的技术以及图林（Turyn）关于不存在的工作，克雷默（Kraemer）完全解决了阿贝尔2组中差异集的存在。尽管在确定Hadamard差集的存在方面已进行了大量工作，但仍有许多问题尚待解决。例如，可逆差异集的存在对许多群体都是开放的。而且，最近在DRAD差异集上的工作已经完成。对于DRAD差异集，许多问题仍未解决。；为了解决某些群体中的存在性问题，我们开发了构造差异集的理论。该理论使用特征，傅立叶分析，伽罗瓦理论和代数数论的方面。我们使用这些工具来创建平铺结构，以产生C2r + 2 x C2r组中的Hadamard差集。还创建了产生可逆和DRAD差异集的平铺结构，组C2 r x C2r。这些组中差异集的存在是先前已知的，但是可以通过不同的方法来显示。使用相同的理论，我们在先前未知的DRAD差异集上建立不存在结果，并在一组576阶中建立差异集不存在。在上一章中，我们给出了两个扩展构造，它们解释了以下所有差异集。订单组36。
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