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
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_325717 |
| Date | January 2006 |
| Contributors | Yeung, Wai Ling Winnie., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | print, v, 100 leaves ; 30 cm. |
| Rights | Use 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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