文档介绍：Codes and Curves
Judy L. Walker
Author address:
Department of Mathematics and Statistics, University
of Nebraska, Lincoln, NE 68588-0323
E-mail address: ******@
1991 Mathematics Subject Classification. Primary 11T71, 94B27;
Secondary 11D45, 11G20, 14H50, 94B05, 94B65.
The author was supported in part by NSF Grant #DMS-9709388.
Contents
IAS/Park City Mathematics Institute ix
Preface xi
Chapter 1. Introduction to Coding Theory 1
. Overview 1
§
. Cyclic Codes 6
§
Chapter 2. Bounds on Codes 9
. Bounds 9
§
. Asymptotic Bounds 12
§
Chapter 3. Algebraic Curves 17
. Algebraically Closed Fields 17
§
. Curves and the Projective Plane 18
§
Chapter 4. Nonsingularity and the Genus 23
. Nonsingularity 23
§
. 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
. Groups 45
§
. Rings, Fields, Ideals, and Factor Rings 46
§
. Vector Spaces 51
§
. Homomorphisms and Isomorphisms 52
§
Appendix B. Finite Fields 55
. Background and Terminology 55
§
. Classi
cation of Finite Fields 56
§
. Optional Exercises 59
§
Appendix C. Projects 61
. Dual Codes and Parity Check Matrices 61
§
. BCH Codes 61
§
. Hamming Codes 62
§
. Golay Codes 62
§
. MDS Codes 62
§
. 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
17-27, 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 t