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Unformatted text preview: Codes and Curves Judy L. Walker Author address: Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 685880323 Email address : jwalker@math.unl.edu 1991 Mathematics Subject Classification. Primary 11T71, 94B27; Secondary 11D45, 11G20, 14H50, 94B05, 94B65. The author was supported in part by NSF Grant #DMS9709388. Contents IAS/Park City Mathematics Institute ix Preface xi Chapter 1. Introduction to Coding Theory 1 1.1. Overview 1 1.2. Cyclic Codes 6 Chapter 2. Bounds on Codes 9 2.1. Bounds 9 2.2. Asymptotic Bounds 12 Chapter 3. Algebraic Curves 17 3.1. Algebraically Closed Fields 17 3.2. Curves and the Projective Plane 18 Chapter 4. Nonsingularity and the Genus 23 4.1. Nonsingularity 23 4.2. Genus 26 vii viii Contents Chapter 5. Points, Functions, and Divisors on Curves 29 Chapter 6. Algebraic Geometry Codes 37 Chapter 7. Good Codes from Algebraic Geometry 41 Appendix A. Abstract Algebra Review 45 A.1. Groups 45 A.2. Rings, Fields, Ideals, and Factor Rings 46 A.3. Vector Spaces 51 A.4. Homomorphisms and Isomorphisms 52 Appendix B. Finite Fields 55 B.1. Background and Terminology 55 B.2. Classification of Finite Fields 56 B.3. Optional Exercises 59 Appendix C. Projects 61 C.1. Dual Codes and Parity Check Matrices 61 C.2. BCH Codes 61 C.3. Hamming Codes 62 C.4. Golay Codes 62 C.5. MDS Codes 62 C.6. Nonlinear Codes 62 Bibliography 65 IAS/Park City Mathematics Institute AMS will insert this ix Preface These notes summarize a series of lectures I gave as part of the IAS/PCMI Mentoring Program for Women in Mathematics, held May 1727, 1999 at the Institute for Advanced Study in Princeton, NJ with funding from the National Science Foundation. The material included is not original, but the exposition is new. The booklet [ LG ] also contains an introduction to algebraic geometric coding theory, but its intended audience is researchers specializing in either coding theory or algebraic geometry and wanting to understand the connec tions between the two subjects. These notes, on the other hand, are designed for a general mathematical audience. In fact, the lectures were originally designed for undergraduates. I have tried to retain the conversational tone of the lectures, and I hope that the reader will find this monograph both accessible and useful. Exercises are scattered throughout, and the reader is strongly encouraged to work through them. Of the sources listed in the bibliography, it should be pointed out that [ CLO2 ], [ Ga ], [ H ], [ L ], [ MS ], [ NZM ] and [ S ] were used most intensively in preparing these notes. In particular: Theorem 1.11, which gives some important properties of cyclic codes, can be found in [ MS ]....
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