Algebraic curves and applications to coding theory.

by Yan Cho Hung. ; Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. ; Includes bibliographical references (leaves 122-124). ; Abstract also in Chinese. ; Chapter 1 --- Complex algebraic curves --- p.6 ; Chapter 1.1 --- Foundations --- p.6 ; Chapter 1.1.1 --- Hilbert Nullstellensatz --- p.6 ; Chapter 1.1.2 --- Complex algebraic curves in C2 --- p.9 ; Chapter 1.1.3 --- Complex projective curves in P2 --- p.11 ; Chapter 1.1.4 --- Affine and projective curves --- p.13 ; Chapter 1.2 --- Algebraic properties of complex projective curves in P2 --- p.16 ; Chapter 1.2.1 --- Intersection multiplicity --- p.16 ; Chapter 1.2.2 --- Bezout's theorem and its applications --- p.18 ; Chapter 1.2.3 --- Cubic curves --- p.21 ; Chapter 1.3 --- Topological properties of complex projective curves in P2 --- p.23 ; Chapter 1.4 --- Riemann surfaces --- p.26 ; Chapter 1.4.1 --- Weierstrass &-function --- p.26 ; Chapter 1.4.2 --- Riemann surfaces and examples --- p.27 ; Chapter 1.5 --- Differentials on Riemann surfaces --- p.28 ; Chapter 1.5.1 --- Holomorphic differentials --- p.28 ; Chapter 1.5.2 --- Abel's Theorem for tori --- p.31 ; Chapter 1.5.3 --- The Riemann-Roch theorem --- p.32 ; Chapter 1.6 --- Singular curves --- p.36 ; Chapter 1.6.1 --- Resolution of singularities --- p.37 ; Chapter 1.6.2 --- The topology of singular curves --- p.45 ; Chapter 2 --- Coding theory --- p.48 ; Chapter 2.1 --- An introduction to codes --- p.48 ; Chapter 2.1.1 --- Efficient noiseless coding --- p.51 ; Chapter 2.1.2 --- The main coding theory problem --- p.56 ; Chapter 2.2 --- Linear codes --- p.58 ; Chapter 2.2.1 --- Syndrome decoding --- p.63 ; Chapter 2.2.2 --- Equivalence of codes --- p.65 ; Chapter 2.2.3 --- An introduction to cyclic codes --- p.67 ; Chapter 2.3 --- Special linear codes --- p.71 ; Chapter 2.3.1 --- Hamming codes --- p.71 ; Chapter 2.3.2 --- Simplex codes --- p.72 ; Chapter 2.3.3 --- Reed-Muller codes --- p.73 ; Chapter 2.3.4 --- BCH codes --- p.75 ; Chapter 2.4 --- Bounds on codes --- p.77 ; Chapter 2.4.1 --- Spheres ...