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dc.contributor.advisorFlannery, Dane
dc.contributor.authorEgan, Ronan
dc.description.abstractThis thesis is a compilation of results dealing with cocyclic development of pairwise combinatorial designs. Motivated by a classification of the indexing and extension groups of the Paley Hadamard matrices due to de Launey and Stafford, we investigate cocyclic development of the so-called generalized Sylvester (or Drake) Hadamard matrices. We describe the automorphism groups and derive strict conditions on possible indexing groups, addressing research problems of de Launey and Flannery in doing so. The shift action, discovered by Horadam, is a certain action of any finite group on the set of its 2-cocycles with trivial coefficients, which preserves both cohomological equivalence and orthogonality. We answer questions posed by Horadam about the shift action, in particular regarding its fixed points. One of our main innovations is the concept of linear shift representation. We give an algorithm for calculating the matrix group representation of a shift action, which enables us to compute with the action in a natural setting. We prove detailed results on reducibility, and discuss the outcomes of some computational experiments, including searches for orthogonal cocycles. Using the algorithms developed for shift representations, and other methods, we classify up to equivalence all cocyclic BH(n,p)s where p is an odd prime (necessarily dividing n) and np < 100. This was achievable with the further aid of our new non-existence results for a wide range of orders.en_IE
dc.subjectAlgebraic design theoryen_IE
dc.subjectCocyclic developmenten_IE
dc.subjectPairwise combinatorial designsen_IE
dc.subjectSchool of Mathematics, Statistics and Applied Mathematicsen_IE
dc.titleTopics in cocyclic development of pairwise combinatorial designsen_IE
dc.contributor.funderNational University of Ireland Hardiman Scholarshipen_IE
dc.contributor.funderIrish Research Councilen_IE
dc.local.noteThis thesis focuses on matrices that have some special combinatorial properties, which have applications to a variety of areas such as cryptography and quantum information theory. We study the underlying algebraic structure of these matrices, and develop methods to construct them.en_IE

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