Absolutely irreducible curves with applications to combinatorics and coding theory

Citation

McGuire, Gary M. (1995) Absolutely irreducible curves with applications to combinatorics and coding theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1CZ2-6S05. https://resolver.caltech.edu/CaltechETD:etd-10122007-094935

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We investigate some problems in algebraic coding theory and finite geometry by relating them to polynomials in two variables and applying Weil's theorem. We prove absolute irreducibility of polynomials arising in this way using Bezout's theorem. In Chapter 2 we investigate certain cyclic codes, and we show that there are codewords of a certain weight by proving that some polynomials are absolutely irreducible and applying Weil's theorem. In Chapter 3 we investigate the existence of hyperovals which have the form [...] in finite projective planes of even order, and we show that there must be three collinear points by proving that some polynomials are absolutely irreducible and applying Weil's theorem. In Chapter 4 we discuss Galois rings of order [...]. We construct a relative difference set from these, and hence an affine plane, which we prove is Desarguesian. We also construct binary codes from the Galois rings, and we prove that there are codewords of a certain weight in the natural generalization of the Preparata and Goethals codes by proving that some polynomials are absolutely irreducible and applying Weil's theorem.

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