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Algebraic Geometric Codes: Basic Notions Mathematical Surveys and Monographs http://dx.doi.org/10.1090/surv/139 Algebraic Geometric Codes: Basic Notions Mathematical Surveys and Monographs Volume 139 Algebraic Geometric Codes: Basic Notions Michael Tsfasman Serge Vladut Dmitry Nogin American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair Our research while working on this book was supported by the French National Scien­ tific Research Center (CNRS), in particular by the Inst it ut de Mathematiques de Luminy and the French-Russian Poncelet Laboratory, by the Institute for Information Transmis­ sion Problems, and by the Independent University of Moscow. It was also supported in part by the Russian Foundation for Basic Research, projects 99-01-01204, 02-01-01041, and 02-01-22005, and by the program Jumelage en Mathematiques. 2000 Mathematics Subject Classification. Primary 14Hxx, 94Bxx, 14G15, 11R58; Secondary 11T23, 11T71. For additional information and updates on this book, visit www.ams.org/bookpages/surv-139 Library of Congress Cataloging-in-Publication Data Tsfasman, M. A. (Michael A.), 1954- Algebraic geometry codes : basic notions / Michael Tsfasman, Serge Vladut, Dmitry Nogin. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 139) Includes bibliographical references and index. ISBN 978-0-8218-4306-2 (alk. paper) 1. Coding theory. 2. Number theory. 3. Geometry, Algebraic. I. Vladut, S. G. (Serge G.), 1954- II. Nogin, Dmitry, 1966- III. Title. QA268 .T754 2007 003'.54—dc22 2007061731 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams. org. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. © The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 Contents Preface ix Advice to the Reader xvii Chapter 1. Codes 1 1.1. Codes and Their Parameters 1 1.1.1. Definition of a Code 1 1.1.2. [n,k,d]q Systems 3 1.1.3. Spectra and Duality 7 1.1.4. Bounds 15 1.1.5. Bounds for Higher Weights 21 1.1.6. Duality for Generalized Spectra 29 1.2. Examples and Constructions 33 1.2.1. Codes of Genus Zero 33 1.2.2. Code Families 36 1.2.3. Constructions 44 1.3. Asymptotic Problems 49 1.3.1. Main Asymptotic Problem 49 1.3.2. Asymptotic Bounds 51 1.3.3. Asymptotic Bounds for B er Weights 57 1.3.4. Polynomiality 60 1.3.5. Other Asymptotics 64 Historical and Bibliographic Notes 67 Chapt< er 2. Curves 69 2.1. Algebraic Curves 69 2.1.1. Quasiprojective Varieties 70 2.1.2. Quasiprojective Curves 77 2.1.3. Divisors 79 2.1.4. Jacobians 85 2.1.5. Riemann Surfaces 88 2.2. Riemann-Roch Theorem 91 2.2.1. Differential Forms 91 2.2.2. Riemann-Roch Theorem 95 2.2.3. Hurwitz Formula 100 2.2.4. Special Divisors 102 2.2.5. Cartier Operator 104 2.3. Singular Curves 106 2.3.1. Normalization 106 vi CONTENTS 2.3.2. Double-Point Divisor 107 2.3.3. Plain Curves 108 2.4. Elliptic Curves 111 2.4.1. Group Law 111 2.4.2. Isomorphisms and the j-invariant 114 2.4.3. Isogenies 115 2.4.4. Complex Elliptic Curves 118 2.5. Curves over Nonclosed Fields 120 2.5.1. Function Fields 120 2.5.2. Places of a Function Field 122 2.5.3. Divisors 125 2.5.4. Function Fields and Algebraic Curves 126 Historical and Bibliographic Notes 131 Chapter 3. Curves over Finite Fields 133 3.1. Zeta Function 133 3.1.1. Definition and Rationality 134 3.1.2. Functional Equation 138 3.1.3. Weil Theorem and Its Corollaries 141 3.1.4. Explicit Formula 143 3.1.5. Pellikaan's Two-Variable Zeta Function 144 3.2. Asymptotics 146 3.2.1. Drinfeld-Vladut Theorem 146 3.2.2. Lower Asymptotic Bounds 147 3.2.3. Points of Higher Degrees 147 3.2.4. Asymptotics for the Jacobian 149 3.2.5. Asymptotically Exact Families 151 3.3. Elliptic Curves over Finite Fields 158 3.3.1. Isomorphism Classes 158 3.3.2. Isogeny Classes 161 3.3.3. Endomorphism Ring and the Zeta Function 162 3.3.4. Structure of E(¥q) 163 3.4. Remarkable Examples 165 3.4.1. Hermitian, Sub-Hermitian, and Maximal Curves 165 3.4.2. Kummer and Artin-Schreier Covers 167 3.4.3. Garcia-Stichtenoth Towers 177 3.4.4. Curves of Small Genera 178 3.5. Connection with Exponential Sums 180 3.5.1. Number of Points on Fermat Curves 180 3.5.2. L-functions of Characters 181 3.5.3. Estimates for Exponential Sums 183 Historical and Bibliographic Notes 187 Chapter 4. Algebraic Geometry Codes 191 4.1. Constructions and Properties 191 4.1.1. Basic Algebraic Geometry Constructions and Their Parameters 192 4.1.2. Duality and Spectra 198 4.1.3. Decoding Problem 202 CONTENTS vii 4.2. Additional Bounds and Constructions 206 4.2.1. Extra Bounds 206 4.2.2. Variants of the Basic Construction 216 4.2.3. Partial Algebraic Geometry Codes 219 4.3. Characterization of Algebraic Geometry Codes 225 4.3.1. Three AG Levels 225 4.3.2. All Linear Codes Are Weakly AG 226 4.3.3. Criteria 228 4.4. Examples 232 4.4.1. Codes of Small Genera 232 4.4.2. Elliptic Codes 234 4.4.3. Hermitian Codes 241 4.4.4. Other Examples 245 4.4.5. Generalized Algebraic Geometry Codes 247 4.5. Asymptotic Results 250 4.5.1. The Basic Algebraic Geometry Bound and Its Variants 250 4.5.2. Expurgation Bound and Codes with Many Light Vectors 253 4.5.3. Constructive Bounds 263 4.5.4. Other Bounds 266 4.6. Nonlinear Algebraic Geometry Constructions 271 4.6.1. Elkies Codes 271 4.6.2. Xing Codes 280 Historical and Bibliographic Notes 284 Appendix. Summary of Results and Tables 287 A.l. Codes of Finite Length 287 A. 1.1. Bounds 287 A. 1.2. Parameters of Some Codes 288 A. 1.3. Parameters of Some Constructions 289 A.2. Asymptotic Bounds 292 A.2.1. List of Bounds 292 A.2.2. Comparison Diagrams 295 A.2.3. Behaviour at the Endpoints 299 A.2.4. Numerical Values 299 A.3. Additional Bounds 306 A.3.1. Constant-Weight Codes 306 A.3.2. Self-dual Codes 306 A.3.3. Bounds for Higher Weights 307 Bibliography 309 List of Names 329 Index 331 Preface The book you hold in your hands is devoted to algebraic geometry codes, a com­ paratively young domain which emerged in the early 1980s at the meeting-point of several fields of mathematics. On the one side we see such respectable, well- . developed, and difficult areas as algebraic geometry and algebraic number theory; on the other is a twentieth-century creation, information transmission theory (with its algebraic daughter, the theory of error-correcting codes), as well as combina­ torics, finite geometries, dense sphere packings, and so on. The relation between the two groups of domains, a priori distant from each other, was discovered by Valery Goppa. At the beginning of 1981 he presented his discovery at the algebra seminar of the Moscow State University; among the listeners there was Yuri Manin, who (being maybe the only one to understand), in turn, related the story at his seminar. One of the authors of this book (Tsfasman) was there and had the first chance in his life to hear the words error-correcting code. A couple of months later he, another author (Vladut;), and Thomas Zink wrote a small paper improving the asymptotic Gilbert-Varshamov bound. To the great astonishment of the authors, who were in doubt whether it was worth publishing or not, the world mathematical community, represented by both coding theorists and algebraic geometers, expressed lively interest. (All of a sudden, the authors found out that for a quarter of a century the asymptotic Gilbert-Varshamov bound had been attacked in vain, which had led to a widely spread—though not unanimous— opinion of its tightness.) This gave birth to a new domain, the algebraic geometry codes. We (Tsfasman and Vladut;) were asked to write a book that could be of use to both coding specialists and algebraic geometers. The book [TV91] was published in 1991 by Kluwer Academic Publishers, and the Russian edition was scheduled and prepared for publication the same year by a Russian publisher, Nauka. The destiny of Russia was, however, different, which we by no means regret. The next time we were asked to publish the book in Russian came ten years later.
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