Using cohomology we show that in studying the existence of an abelian non-splitting (4t, 2, 4t, 2t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets.
This thesis applies the two-dimensional cohomology of finite groups and the theory of cocycles to two areas: combinatorics and error correcting codes.In the first case, it is shown that several combinatorial objects denned on groups: semi-regular central relative difference sets, sequences with certain auto-correlation properties and cocyclic generalized Hadamard matrices can all be thought of as being equivalent to a single underlying concept, namely, a base sequence. Using this equivalence we can move from one of these objects to another, and so use whichever formulation is most useful. This gives a unified way of studying the combinatorial properties of extension groups in both the splitting and, more difficult, non-splitting cases. General results proven about base sequences imply corresponding results about the equivalent combinatorial objects. We apply this in two cases, generalized perfect binary arrays and perfect quaternary arrays and show, for example, that these objects are equivalent to Hadamard matrices of a specific easy to describe form. We also prove, using cohomology and generalized perfect binary arrays, that in studying the existence of abelian non-splitting semi-regular difference sets (relative to order two subgroups) we can assume the groups in question have a simple "canonical" form. We also show that a perfect quaternary array is explicitly equivalent to a particular type of generalized perfect binary array.In the theory of error correcting codes, cohomology allows us to give an isomorphism between a group ring code and the direct sum of other group ring codes that have been "twisted" by cocycles. This is a known result in certain cases (for example in the abelian case it is a generalized Chinese Remainder Theorem) but our approach gives the isomorphism explicitly in terms of multiplication by a Vandermonde matrix.We use the isomorphism to construct new codes from known "smaller" codes. For example, over a field of odd characteristic, we have &u + v\u-v code construction which has a distance estimated no worse than the well known u \ u + v construction. It has an advantage over that construction, however, because when given a cyclic code
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