The Brill-Noether theorem with applications to A. G. Goppa codes and exponential sums.
Item
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Title
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The Brill-Noether theorem with applications to A. G. Goppa codes and exponential sums.
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Identifier
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AAI9207110
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identifier
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9207110
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Creator
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Polemis, Despina.
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Contributor
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Adviser: Carlos J. Moreno
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Date
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1991
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Throughout this dissertation we are primarily concerned with various algebraic and geometric properties of curves in relation to coding theory and the theory of exponential sums.;The curves of our interest are defined over finite fields which leads us to reconsider well known theorems as for instance the Brill-Noether theorem.;The problem of resolution of singularities lead us to investigate the selected methods available in order to desingularize a plane singular curve defined over a finite field. We present algorithmic models which describe these methods and we estimate their time complexities. We prove the relations among the exponential sums built from singular curves and the ones built from their equivalents.;The classical concepts of adjoint divisors, adjoint curves and their relation to the Brill-Noether theorem are reconsidered from a constructive point of view in which the field of constants is a finite field. We also compute the discriminant divisor utilizing a geometric and an algebraic method.;The problem of effectively calculating with the Riemann-Roch theorem is also considered in this thesis. We present two algorithmic constructions which estimate a basis for the vector space {dollar}{lcub}\cal L{rcub}(G){dollar}. The first construction is the algorithm EVG based on the Cremona transformations and the second is the algorithm MBR based on the normalization process. We estimate the time complexity of both algorithms. They both serve as basic tools for the construction of algebraic geometric Goppa codes evolved by singular curves.;We give a new bound on the minimum distance of the dual of a Goppa binary subfield subcode generated from a singular curve. We obtain the new bound using exponential sum techniques.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
