by Ho Kin Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 68-69). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Introduction to Coding Theory --- p.9 / Chapter 1.1 --- Definition of a code --- p.10 / Chapter 1.2 --- Maximum Likelihood Decoding --- p.11 / Chapter 1.3 --- Syndrome Decoding --- p.12 / Chapter 1.4 --- Two Kinds of Errors and Concatenated Code --- p.14 / Chapter 2 --- Basic Knowledge of Algebraic Curve --- p.16 / Chapter 2.1 --- Affine and Projective Curve --- p.16 / Chapter 2.2 --- Regular Functions and Maps --- p.17 / Chapter 2.3 --- Divisors and Differential forms --- p.19 / Chapter 2.4 --- Riemann-Roch Theorem --- p.21 / Chapter 3 --- Construction of Algebraic Geometric Code --- p.23 / Chapter 3.1 --- L-construction --- p.23 / Chapter 3.2 --- Ω -construction --- p.24 / Chapter 3.3 --- Duality --- p.26 / Chapter 4 --- Basic Error Processing --- p.28 / Chapter 4.1 --- Error Locators and Syndromes --- p.28 / Chapter 4.2 --- Finding an Error Locator --- p.29 / Chapter 5 --- Full Error Processing for Code on Curve of Genus1 --- p.34 / Chapter 5.1 --- Syndrome table --- p.34 / Chapter 5.2 --- Syndrome table --- p.36 / Chapter 5.3 --- The algorithm of Full Error Processing --- p.38 / Chapter 5.4 --- A simple Example --- p.40 / Chapter 6 --- General Full Error Processing --- p.47 / Chapter 6.1 --- Row Candidate and Column Candidate --- p.47 / Chapter 6.2 --- Consistency --- p.49 / Chapter 6.3 --- Majority voting --- p.50 / Chapter 6.4 --- Example --- p.53 / Chapter 7 --- Application of Algebraic Geometric Code --- p.60 / Chapter 7.1 --- Communication --- p.60 / Chapter 7.2 --- Cryptosystem --- p.62 / Bibliography

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 in Zn --- p.77 / Chapter 2.4.2 --- Perfect codes --- p.78 / Chapter 2.4.3 --- Famous numbers Ar (n，d) and the sphere-covering and sphere packing bounds --- p.79 / Chapter 2.4.4 --- The Singleton and Plotkin bounds --- p.81 / Chapter 2.4.5 --- The Gilbert-Varshamov bound --- p.83 / Chapter 3 --- Algebraic curves over finite fields and the Goppa codes --- p.85 / Chapter 3.1 --- Algebraic curves over finite fields --- p.85 / Chapter 3.1.1 --- Affine varieties --- p.85 / Chapter 3.1.2 --- Projective varieties --- p.37 / Chapter 3.1.3 --- Morphisms --- p.89 / Chapter 3.1.4 --- Rational maps --- p.91 / Chapter 3.1.5 --- Non-singular varieties --- p.92 / Chapter 3.1.6 --- Smooth models of algebraic curves --- p.93 / Chapter 3.2 --- Goppa codes --- p.96 / Chapter 3.2.1 --- Elementary Goppa codes --- p.96 / Chapter 3.2.2 --- The affine and projective lines --- p.98 / Chapter 3.2.3 --- Goppa codes on the projective line --- p.102 / Chapter 3.2.4 --- Differentials and divisors --- p.105 / Chapter 3.2.5 --- Algebraic geometric codes --- p.112 / Chapter 3.2.6 --- Codes with better rates than the Varshamov- Gilbert bound and calculation of parameters --- p.116 / Bibliography

Kwok Pui Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 66-67). / Abstracts in English and Chinese. / Abstract --- p.i / Abstract (Chinese Version) --- p.ii / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Linear Network Coding --- p.7 / Chapter 2.1 --- Single Source Network Coding --- p.8 / Chapter 2.2 --- Descriptions of Linear Network Codes --- p.9 / Chapter 2.3 --- Desirable Properties of Linear Network Codes --- p.12 / Chapter 2.4 --- Linear Network Codes Constructions --- p.14 / Chapter 3 --- Node-based Characterization --- p.16 / Chapter 3.1 --- Channel-based characterization --- p.16 / Chapter 3.2 --- A Necessary Condition for the Existence of Linear Network Codes --- p.17 / Chapter 3.3 --- Insufficiency of the condition --- p.22 / Chapter 4 --- Relation between Linear Network Codes --- p.25 / Chapter 4.1 --- Relation between Multicast and Broadcast --- p.26 / Chapter 4.1.1 --- Auxiliary Graph --- p.26 / Chapter 4.2 --- Relation between Broadcast and Dispersion --- p.29 / Chapter 4.2.1 --- Expanded Graph --- p.29 / Chapter 4.3 --- Relation between Dispersion and Generic Net- work Code --- p.31 / Chapter 4.3.1 --- Edge Disjoint Path --- p.31 / Chapter 4.3.2 --- Path Rearrangement --- p.34 / Chapter 4.3.3 --- Extended Graph --- p.50 / Chapter 5 --- Upper Bound on the Size of the Base Field --- p.57 / Chapter 5.1 --- Base Field Size Requirement --- p.58 / Chapter 5.1.1 --- Linear Multicast --- p.58 / Chapter 5.1.2 --- Linear Broadcast --- p.58 / Chapter 5.1.3 --- Linear Dispersion --- p.59 / Chapter 5.1.4 --- Generic Network Code --- p.60 / Chapter 5.2 --- Upper Bounds Comparison for Generic Network Code --- p.61 / Chapter 6 --- Future Work --- p.62 / Bibliography --- p.66

In 1998, J. P. Hansen introduced the construction of an error-correcting code over a finite field Fq from a convex integral polytope in R2. Given a polytope P ⊂ R2, there is an associated toric variety XP , and Hansen used the cohomology and intersection theory of divisors on XP to determine explicit formulas for the dimension and minimum distance of the associated toric code CP . We begin by reviewing the basics of algebraic coding theory and toric varieties and discuss how these areas intertwine with discrete geometry. Our first results characterize certain polygons that generate and do not generate maximum distance separable (MDS) codes and Almost-MDS codes. In 2006, Little and Schenck gave formulas for the minimum distance of certain toric codes corresponding to smooth toric surfaces with rank(Pic(X)) = 2 and rank(Pic(X)) = 3. Additionally, they gave upper and lower bounds on the minimum distance of an arbitrary toric code CP by finding a subpolygon of P with a maximal, nontrivial Minkowski sum decomposition. Following this example, we give explicit formulas for the minimum distance of toric codes associated with two families of smooth toric threefolds with rank(Pic(X)) = 2, characterized by G. Ewald and A. Schmeinck in 1993. Lastly, we give explicit formulas for the dimension of a toric code generated from a Minkowski sum of a finite number of polytopes in R2 and R3 and a lower bound for the minimum distance.

No description available.

Thesis (Ph. D.)--University of Washington, 2008. / Vita. Includes bibliographical references (p. 73-78).

Thesis (Ph.D.)--University of Texas at Arlington, 2008.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2002. / Includes bibliographical references (leaves 105-110). Also available in electronic version. Access restricted to campus users.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 67-68). Also available in electronic version. Access restricted to campus users.

Thesis (M. S.)--Computing, Georgia Institute of Technology, 2009. / Committee Chair: Pande, Santosh; Committee Member: Clark, Nate; Committee Member: Yalamanchili, Sudhakar. Part of the SMARTech Electronic Thesis and Dissertation Collection.