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.1Chapter 2 --- Finite Fields --- p.4Chapter 2.1 --- Basic Properties of Finite Fields --- p.4Chapter 2.2 --- Existence and Uniqueness of Finite Fields --- p.8Chapter 2.3 --- Algorithms in Factoring Polynomials --- p.11Chapter 2.3.1 --- Factorization of xn ´ؤ 1 --- p.11Chapter 2.3.2 --- Berlekamp Algorithm for Factorizing an Arbitrary Polynomial --- p.13Chapter 3 --- Algebraic Curves --- p.17Chapter 3.1 --- Affine and Projective Curves --- p.17Chapter 3.2 --- Local Properties and Intersections of Curves --- p.19Chapter 3.3 --- Linear Systems of Curves and Noether's Theorem --- p.24Chapter 3.4 --- Rational Function and Divisors --- p.29Chapter 3.5 --- Differentials on a Curve --- p.34Chapter 3.6 --- Riemann-Roch Theorem --- p.36Chapter 4 --- Coding Theory --- p.46Chapter 4.1 --- Introduction to Coding Theory --- p.46Chapter 4.1.1 --- Basic Definitions for Error-Correcting Code --- p.46Chapter 4.1.2 --- Geometric Approach to Error-Correcting Capabilities of Codes --- p.48Chapter 4.2 --- Linear Codes --- p.49Chapter 4.2.1 --- The Dual of a Linear Code --- p.54Chapter 4.2.2 --- Syndrome Decoding --- p.57Chapter 4.2.3 --- Extension of Basic Field --- p.60Chapter 4.3 --- The Main Problem in Coding Theory --- p.62Chapter 4.3.1 --- "Elementary Results on Aq(n, d)" --- p.63Chapter 4.3.2 --- "Lower Bounds on Aq(n, d)" --- p.63Chapter 4.3.3 --- "Upper Bounds on Aq(n,d)" --- p.65Chapter 4.3.4 --- Asymptotic Bounds --- p.67Chapter 4.4 --- Rational Codes --- p.68Chapter 4.4.1 --- Hamming Codes --- p.68Chapter 4.4.2 --- Codes on an Oval --- p.69Chapter 4.4.3 --- Codes on a Twisted Cubic Curve --- p.78Chapter 4.4.4 --- Normal Rational Codes --- p.82Chapter 4.5 --- Goppa Codes --- p.84Chapter 4.5.1 --- Classical Goppa Codes --- p.85Chapter 4.5.2 --- Geometric Goppa Codes --- p.88Chapter 4.5.3 --- Good Codes from Algebraic Geometry --- p.91Chapter 4.6 --- A Recent Non-linear Code Improving the Tsfasman- Vladut-Zink Bound --- p.93Bibliography --- p.9