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Different Aspects of Coding Theory, Volume 50 http://dx.doi.org/10.1090/psapm/050 Other Titles in This Series 50 Robert Calderbank, editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994) 48 Walter Gautschi, editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and William Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R. W. McKelvey, editor, Environmental and natural resource mathematics (Eugene, Oregon, August 1984) 31 B. Gopinath, editor, Computer communications (Denver, Colorado, January 1983) 30 Simon A. Levin, editor, Population biology (Albany, New York, August 1983) 29 R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981) 28 R. Gnanadesikan, editor, Statistical data analysis (Toronto, Ontario, August 1982) 27 L. A. Shepp, editor, Computed tomography (Cincinnati, Ohio, January 1982) 26 S. A. Burr, editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981) 25 S. I. Gass, editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979) 24 W. F. Lucas, editor, Game theory and its applications (Biloxi, Mississippi, January 1979) 23 R. V. Hogg, editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980) 22 G. H. Golub and J. Oliger, editors, Numerical analysis (Atlanta, Georgia, January 1978) 21 P. D. Lax, editor, Mathematical aspects of production and distribution of energy (San Antonio, Texas, January 1976) (Continued in the back of this publication) Proceedings of Symposia in APPLIED MATHEMATICS Volume 50 Different Aspects of Coding Theory American Mathematical Society Short Course January 2-3, 1995 San Francisco, California Robert Calderbank Editor American Mathematical Society !f Providence, Rhode Island ^VDED v* LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE CODING THEORY HELD IN SAN FRANCISCO, CALIFORNIA JANUARY 2-3, 1995 The AMS Short Course Series is sponsored by the Society's Program Committee on National Meetings. The Series is under the direction of the Short Course Subcommittee of the Program Committee for National Meetings. 1991 Mathematics Subject Classification. Primary 62K05, 68Q25, 68Q68, 93B15, 94A05* Secondary 05B25, 51A35, 51A40, 58F03, 68P25, 68Q15, 94A14, 94A40, 94A60, 94B12, 94B27. Library of Congress Cataloging-in-Publication Data Different aspects of coding theory : American Mathematical Society short course, January 2-3, 1995, San Francisco, California / Robert Calderbank, editor. p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 50) Includes bibliographical references and index. ISBN 0-8218-0379-4 1. Coding theory—Congresses. I. Calderbank, Robert, 1954- . II. Series. QA268.D54 1995 003,.54—dc20 95-35165 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQmath. ams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1995 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. Printed on recycled paper. 10 9 8 7 6 5 4 3 2 1 00 99 98 97 96 95 Table of Contents Preface vii Coding Theory as Discrete Applied Mathematics A. R. CALDERBANK Modulation Codes for Digital Data Storage BRIAN MARCUS, RON M. ROTH AND PAUL H. SIEGEL 41 Symbolic Dynamics and Connections to Coding Theory, Automata Theory and System Theory BRIAN MARCUS 95 Multilingual Dictionary DAVE FORNEY, BRIAN MARCUS, N.T. SINDHUSHAYANA AND MITCHELL TROTT 109 Algebraic Geometric Codes HENNING STICHTENOTH 139 Codes, Quadratic Forms and Finite Geometries WILLIAM M. KANTOR 153 Codes (Spherical) and Designs (Experimental) R. H. HARDIN AND N. J. A. SLOANE 179 The Use of Coding Theory in Computational Complexity JOAN FEIGENBAUM 207 Preface The lectures at the Short Course in San Francisco and the different chapters of this book emphasize connections between coding theory and coding practice, and between coding theory and different parts of mathematics. The principal purpose of coding theory is the reliable transmission or storage of data. The introductory chapter provides the basics of algebraic coding theory, and the design of codes for the Gaussian channel. It is intended to set the stage for the contributions that follow. The chapter by Marcus, Roth and Siegel focuses on mag• netic recording, where binary data written on a disk is required to satisfy certain constraints. The encoding of random data as constrained sequences is accomplished by means of a finite state machine. Shannon, in 1948, introduced the notion of ca• pacity; the maximum data rate supported by the universe of constrained sequences. However methods of encoding and decoding at or very close to this achievable rate were developed more recently through connections with symbolic dynamics. Here we also see connections with finite automata since the constrained sequences can be thought of as a language that is accepted by the encoder. In fact many basic concepts have grown up independently in coding theory, linear systems theory and automata theory. The multilingual dictionary compiled by Forney, Marcus, Sind- hushayana and Trott connects these different worlds and how they have evolved in parallel. One of the objectives of the Short Course and this book was to promote more crosspollination between these different mathematical communities. One of the most interesting connections between coding theory and other parts of mathematics is the use of algebraic geometry to construct efficient error- correcting codes. The contribution by Stichtenoth relates these geometric Goppa codes to the polynomial codes constructed by Reed and Solomon that are now found everywhere from computer hard disks to compact disk players. Reed-Solomon and algebraic geometry codes are particular examples of linear codes, but it is sometimes the case that nonlinear codes are more efficient. Examples include the Kerdock and Preparata codes which are closely related to quadratic forms. The chapter by Kantor connects these codes with orthogonal and symplectic geometry, and with extremal families of lines in Euclidean space. A different geometric perspective is evident in the chapter by Hardin and Sloane. Here we are concerned with approximating the properties of a large uni• verse (such as the surface of a sphere) with a small ensemble or design (such as a discrete subset of points). This connects with the construction of experimental designs in statistics, which are used to optimize industrial processes. However there are also fascinating connections with Waring's Problem in number theory. Vll Vlll PREFACE The last chapter by Feigenbaum explores the use of codes in theoretical com• puter science. Here codes are used to improve algorithmic efficiency, in program testing and correction,
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