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Finite fields, algebraic curves and coding theory. / Finite fields, algebraic curves & coding theory

Yeung Wai Ling Winnie. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 99-100). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Finite Fields --- p.4 / Chapter 2.1 --- Basic Properties of Finite Fields --- p.4 / Chapter 2.2 --- Existence and Uniqueness of Finite Fields --- p.8 / Chapter 2.3 --- Algorithms in Factoring Polynomials --- p.11 / Chapter 2.3.1 --- Factorization of xn ´ؤ 1 --- p.11 / Chapter 2.3.2 --- Berlekamp Algorithm for Factorizing an Arbitrary Polynomial --- p.13 / Chapter 3 --- Algebraic Curves --- p.17 / Chapter 3.1 --- Affine and Projective Curves --- p.17 / Chapter 3.2 --- Local Properties and Intersections of Curves --- p.19 / Chapter 3.3 --- Linear Systems of Curves and Noether's Theorem --- p.24 / Chapter 3.4 --- Rational Function and Divisors --- p.29 / Chapter 3.5 --- Differentials on a Curve --- p.34 / Chapter 3.6 --- Riemann-Roch Theorem --- p.36 / Chapter 4 --- Coding Theory --- p.46 / Chapter 4.1 --- Introduction to Coding Theory --- p.46 / Chapter 4.1.1 --- Basic Definitions for Error-Correcting Code --- p.46 / Chapter 4.1.2 --- Geometric Approach to Error-Correcting Capabilities of Codes --- p.48 / Chapter 4.2 --- Linear Codes --- p.49 / Chapter 4.2.1 --- The Dual of a Linear Code --- p.54 / Chapter 4.2.2 --- Syndrome Decoding --- p.57 / Chapter 4.2.3 --- Extension of Basic Field --- p.60 / Chapter 4.3 --- The Main Problem in Coding Theory --- p.62 / Chapter 4.3.1 --- "Elementary Results on Aq(n, d)" --- p.63 / Chapter 4.3.2 --- "Lower Bounds on Aq(n, d)" --- p.63 / Chapter 4.3.3 --- "Upper Bounds on Aq(n,d)" --- p.65 / Chapter 4.3.4 --- Asymptotic Bounds --- p.67 / Chapter 4.4 --- Rational Codes --- p.68 / Chapter 4.4.1 --- Hamming Codes --- p.68 / Chapter 4.4.2 --- Codes on an Oval --- p.69 / Chapter 4.4.3 --- Codes on a Twisted Cubic Curve --- p.78 / Chapter 4.4.4 --- Normal Rational Codes --- p.82 / Chapter 4.5 --- Goppa Codes --- p.84 / Chapter 4.5.1 --- Classical Goppa Codes --- p.85 / Chapter 4.5.2 --- Geometric Goppa Codes --- p.88 / Chapter 4.5.3 --- Good Codes from Algebraic Geometry --- p.91 / Chapter 4.6 --- A Recent Non-linear Code Improving the Tsfasman- Vladut-Zink Bound --- p.93 / Bibliography --- p.99

Identiferoai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_325717
Date January 2006
ContributorsYeung, Wai Ling Winnie., Chinese University of Hong Kong Graduate School. Division of Mathematics.
Source SetsThe Chinese University of Hong Kong
LanguageEnglish, Chinese
Detected LanguageEnglish
TypeText, bibliography
Formatprint, v, 100 leaves ; 30 cm.
RightsUse of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

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