Citation
McGuire, Gary M. (1995) Absolutely irreducible curves with applications to combinatorics and coding theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1CZ26S05. https://resolver.caltech.edu/CaltechETD:etd10122007094935
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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  16 May 1995 
Record Number:  CaltechETD:etd10122007094935 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd10122007094935 
DOI:  10.7907/1CZ26S05 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  4053 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  25 Oct 2007 
Last Modified:  21 Dec 2019 04:07 
Thesis Files

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