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Algebraic Geometry Codes: Advanced Chapters Mathematical Surveys and Monographs Volume 238 Algebraic Geometry Codes: Advanced Chapters Michael Tsfasman Serge VladuÛ Γ Dmitry Nogin 10.1090/surv/238 Mathematical Surveys and Monographs Volume 238 Algebraic Geometry Codes: Advanced Chapters Michael Tsfasman Serge VladuÛ Γ Dmitry Nogin EDITORIAL COMMITTEE Walter Craig Natasa Sesum Robert Guralnick, Chair Benjamin Sudakov Constantin Teleman 2010 Mathematics Subject Classification. Primary 14Hxx, 94Bxx, 14G15, 11R58; Secondary 11R04, 11T71, 11M38, 11R37, 11R42, 14D22, 14J20. For additional information and updates on this book, visit www.ams.org/bookpages/surv-238 Library of Congress Cataloging-in-Publication Data Names: Tsfasman, M. A. (Michael A.), 1954– author. | Vlˇadut¸, S. G. (Serge G.), 1954– author. | Nogin, Dmitry, 1966– author. Title: Algebraic geometry codes: Advanced chapters / Michael Tsfasman, Serge Vlˇadut¸, Dmitry Nogin. Other titles: Title of previous publication: Algebraic geometric codes Description: Providence, Rhode Island: American Mathematical Society, [2019] | Series: Mathe- matical surveys and monographs; volume 238 | Algebraic-geometric codes / by M.A. Tsfasman, published: Dordrecht: Kluwer Academic Publishers, 1991. Republished as: Algebraic geomet- ric codes: Basic notions (Providence, R.I.: American Mathematical Society, 2007). Algebraic geometry codes: Advanced chapters, continues 2007 publication, beginning at chapter 5. | Includes bibliographical references and indexes. Identifiers: LCCN 2019003782 | ISBN 9781470448653 (alk. paper) Subjects: LCSH: Coding theory. | Number theory. | Geometry, Algebraic. | AMS: Algebraic geometry – Curves – Curves. msc | Information and communication, circuits – Theory of error-correcting codes and error-detecting codes – Theory of error-correcting codes and error- detecting codes. msc | Algebraic geometry – Arithmetic problems. Diophantine geometry – Finite ground fields. msc | Number theory – Algebraic number theory: global fields – Arithmetic theory of algebraic function fields. msc | Number theory – Algebraic number theory: global fields – Algebraic numbers; rings of algebraic integers. msc | Number theory – Finite fields and commutative rings (number-theoretic aspects) – Algebraic coding theory; cryptography. msc | Number theory – Zeta and L-functions: analytic theory – Zeta and L- functions in characteristic p.msc| Number theory – Algebraic number theory: global fields – Class field theory. msc | Number theory – Algebraic number theory: global fields – Zeta functions and L-functions of number fields. msc | Algebraic geometry – Families, fibrations – Fine and coarse moduli spaces. msc | Algebraic geometry – Surfaces and higher-dimensional varieties – Arithmetic ground fields. msc Classification: LCC QA268 .T755 2019 | DDC 003/.54–dc23 LC record available at https://lccn.loc.gov/2019003782 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 select pages 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 permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the authors. All rights reserved. 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 https://www.ams.org/ 10987654321 242322212019 Contents Preface vii Chapter 5. Curves with Many Points. I: Modular Curves 1 5.1. Classical Modular Curves 2 5.1.1. Modular Curves 2 5.1.2. Modular Curves over Finite Fields 12 5.1.3. Codes 16 5.2. Drinfeld Modular Curves 21 5.2.1. Elliptic Modules 22 5.2.2. Drinfeld Curves 24 5.2.3. Codes 33 5.3. Elkies Explicit Towers 37 5.3.1. Classical Modular Curve Towers 37 5.3.2. Explicit Towers of Drinfeld Modular Curves 42 Historical and Bibliographic Notes 47 Chapter 6. Class Field Theory 49 6.1. Global Fields 49 6.1.1. Function Fields 50 6.1.2. Number Fields 55 6.2. Local Class Field Theory 62 6.3. Global Class Field Theory 68 6.3.1. Artin Map 68 6.3.2. Main Theorems 71 6.3.3. Explicit Class Field Theory for Function Fields 74 6.4. Class Field Towers 79 6.4.1. Class Field Tower Problem 79 6.4.2. Applications to A(q)81 Historical and Bibliographic Notes 85 Chapter 7. Curves with Many Points. II 87 7.1. Oesterl´e Bound 88 7.2. Deligne–Lusztig Curves 94 7.2.1. Group Codes on Hermitian Curves 94 7.2.2. Suzuki Curves 96 7.2.3. Ree Curves 101 7.2.4. Deligne–Lusztig Curves via Class Field Theory 104 iii iv CONTENTS 7.3. Some Curves of Small Genera 107 7.3.1. Curves with Many Points from Ray Class Fields 107 7.3.2. Fibre Products 111 7.3.3. Maximal Curves Covered by a Hermitian Curve 114 7.4. Recursive Towers 116 7.4.1. Some Basic Facts on Towers 116 7.4.2. Some Specific Garc´ıa–Stichtenoth Towers and the Elkies Conjecture 120 7.4.3. Polynomially Constructible Codes from the W1 Tower 125 7.4.4. A Very Good Tower for q = p2m+1, m ≥ 1 128 7.4.5. Good Recursive Towers over Fq, q ≥ 4 130 7.5. Two Nonstandard Problems 133 7.5.1. Curves with a Prescribed Number of Points 133 7.5.2. Curves for Every Genus 135 Historical and Bibliographic Notes 138 Chapter 8. Infinite Global Fields 141 8.1. Invariants and Basic Inequalities 141 8.1.1. Invariants of Infinite Global Fields 146 8.1.2. Basic Inequalities 149 8.1.3. Limit Zeta Function 155 8.1.4. Limit Explicit Formula 158 8.2. The Generalized Brauer–Siegel Theorem 163 8.2.1. Main Result 163 8.2.2. Bounds for the Brauer–Siegel Ratios 171 8.3. Class Field Tower Examples 175 8.3.1. Unramified Towers with Splitting Conditions 175 8.3.2. Hajir–Maire Tame Towers 183 8.4. Further Theory and Open Questions 185 8.4.1. Common Questions 185 8.4.2. Results Specific for the Function Field Case and Corresponding Problems 188 Historical and Bibliographic Notes 192 Chapter 9. Decoding: Some Examples 195 9.1. List Decoding 195 9.1.1. Johnson Bound 196 9.1.2. Capacity of List Decoding 198 9.2. Guruswami–Sudan Algorithm 200 9.2.1. Decoding of Reed–Solomon Codes 200 9.2.2. Decoding of Algebraic Geometry Codes 203 9.2.3. Representations of Algebraic Geometry Codes 207 9.3. Example: Hermitian Curves 211 9.3.1. Bases and Interpolation 211 9.3.2. Factorization 214 CONTENTS v 9.4. Approaching the Singleton Bound 217 9.4.1. Periodic Affine Subspaces 217 9.4.2. Subspace Designs 218 9.4.3. Case of the General Algebraic Geometry Codes 220 9.4.4. Codes from the W1 Tower 223 Historical and Bibliographic Notes 226 Chapter 10. Sphere Packings 229 10.1. Definitions, Examples, and Constructions 229 10.1.1. Parameters and Some Basic Examples 229 10.1.2. Asymptotic Problems 236 10.1.3. Random Packings 238 10.2. Asymptotically Dense Packings 241 10.2.1. Constructions of Dense Packings 241 10.2.2. Spherical Codes and Kissing Numbers 245 10.2.3. Lattices with Exponentially Large Kissing Numbers 249 10.3. Lattices from Global Fields 253 10.3.1. Additive Constructions 253 10.3.2. Multiplicative Constructions 257 10.3.3. Congruence Constructions 260 10.4. Mordell–Weil Lattices 265 10.4.1. Shioda Lattices 266 10.4.2. Elkies Lattices 270 Appendix: Parameters of Some Packings 275 Historical and Bibliographic Notes 276 Chapter 11. Codes from Multidimensional Varieties 279 11.1. Complete Intersections and Reed–Muller Codes 280 11.1.1. Tsfasman–Serre–Sørensen Bound 280 11.1.2. Generalization to Several Polynomials: Tsfasman– Boguslavsky Conjecture 281 11.1.3. Reed–Muller Codes and the Affine Case 286 11.2. General Algebraic Sets 289 11.2.1. Lachaud’s Bounds 289 11.2.2. Couvreur’s Bound 294 11.3. Codes on Surfaces 301 11.3.1. Some Elements of the Theory of Surfaces 301 11.3.2. Cubic Surfaces over a Finite Field 304 11.3.3. Rational Surfaces for Good Codes 307 11.4. Hermitian Varieties and Quadrics 311 11.4.1. Hermitian Varieties 311 11.4.2. Quadrics 313 11.5. Grassmann and Schubert Codes 319 11.5.1. Grassmann Codes 319 11.5.2. Schubert Codes 323 vi CONTENTS 11.6. Codes from Flag Varieties 328 11.6.1. Flag Varieties 328 11.6.2. Examples 329 11.6.3. Two More Examples 331 Historical and Bibliographic Notes 333 Chapter 12. Applications 335 12.1. Fast Multiplication in Finite Fields 335 12.1.1. Tensor Rank and Bilinear Complexity 335 12.1.2. The Extended Chudnovsky Algorithm 338 12.2. Cryptographic Applications 343 12.2.1. Authentication Codes from Algebraic Curves 343 12.2.2. Arithmetic Secret Sharing Schemes 350 12.3. Quantum Codes 357 12.3.1. Quantum Error-Correcting Codes 357 12.3.2. Quantum Codes from Algebraic Geometry Codes 360 12.3.3. Nonbinary Case 366 12.4. Niederreiter–Rosenbloom–Tsfasman Metric 369 12.4.1. NRT Metric Spaces: Definitions and Bounds 369 12.4.2. Examples and Asymptotic Bounds 372 12.4.3. Uniform Nets and Sequences 378 12.5. Locally Recoverable Codes 384 12.5.1. Optimal LRC Codes 384 12.5.2. Locally Recoverable Codes on Algebraic Curves 386 12.5.3. Asymptotic Behaviour 389 Historical and Bibliographic Notes 395 Some Basic Facts from Volume 1 397 A.1. Codes and Projective Systems 397 A.1.1.
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