sign in sign up
<<
Home , Coding theory, Dave Forney, Robert Calderbank

http://dx.doi.org/10.1090/psapm/050

Other Titles in This Series

50 , editor, Different aspects of coding (San Francisco, California, January 1995) 49 Robert L. Devaney, editor, Complex dynamical systems: The 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 , 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 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, theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the 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, (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 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, : mathematics and models (Duluth, Minnesota, August 1979) 24 W. F. Lucas, editor, and its applications (Biloxi, Mississippi, January 1979) 23 R. V. Hogg, editor, Modern : Methods and applications (San Antonio, Texas, January 1980) 22 G. H. Golub and J. Oliger, editors, (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

Volume 50

Different Aspects of

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 for Storage BRIAN MARCUS, RON M. ROTH AND PAUL H. SIEGEL 41

Symbolic Dynamics and Connections to Coding Theory, and System Theory BRIAN MARCUS 95

Multilingual Dictionary , 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, and to obtain characterizations of complexity classes. It is an opportunity for coding theorists to see familiar constructs at work in a very different context. I would like to use this opportunity to thank all the lecturers at the Short Course, and all the contributors to this volume. I hope that all readers, whether they be coding theory novices or experts in the , will find something to interest them in the different chapters. I would also like to thank Wayne Drady for the time and energy he invested in making the Short Course run smoothly. Finally I would like to thank Susan Pope for creating a coherent whole out of the individual contributions.

A. R. Calderbank AT&T Bell Laboratories Murray Hill, NJ 07974 May 15, 1995 Index (0,1)-RLL constrained system, 55 closed, 112 (0, G/I) constraints, 45 closure, 112 (A, n)-approximate eigenvector, 62 co-deterministic, 56 (5, n)-encoder, 65 , 2, 111, 140, 148, 213:

(m,a)-definite, 56 Z4, 171 (m, a)-sliding block decodable, 70 Z4-Kerdock, 153 (d, fc)-runlength-limited (RLL) constraints, 43 algebraic geometry, 6, 139 (d,fc,s), 43 asymptotically good, 148 (d, k; c) constraints, 44 BCH, 13 x-consistent, 76 binary, 213 e-biased random variables, 215 convolutional, 101, 131 ^-block, 54 constant weight, 185 Z4-valued quadratic form, 172 constrained, 41 coset, 35 action-preserving out-splitting, 103 cyclic, 14 action-preserving parallel-edgification, 103 dual, 3 adjacency matrix, 54, 100, 133 eight-to-fourteen modulation (EFM), 72 afflne plane, 153 finite-constraint length, 116 algebraic function fields, 144 finite-state, 129 alternating bilinear form, 155 finite-to-one sliding block, 127 analog-to-digital converter, 181 generated by a realization, 117, 118 anticipation, 47: generated by a trellis, 119 approximate eigenvector inequality, 63 geometric Goppa, 139 approximation: geometrically uniform, 133 e-approximation , 210 Golay 5, 14 polynomial time approximation scheme, 210 , 132 approximation to sphere, 181 Hamming, 4, 13, 14 finite anticipation, 56 Hermit ian, 141 (local) anticipation, 56 Justesen, 214 (local) co-anticipation, 56 Kerdock, 4, 153 Assmus-Mattson Theorem, 21 linear, 3, 153, 213 band-limited Gaussian channel, 21 linear finite-state, 131 basic out-splitting, 74 linear sliding-block, 132 basic x-consistent partition, 76 modified frequency modulation (MFM), 65 basic x-consistent splitting, 77 modulation, 41 Bezout's Theorem, 142 multiple-spaced run length limited (RLL), binary operation, 159 43 bit-commitment, 217 noncatastrophic, 48 block, 96 noncatastrophic finite-state, 130 block coding theorem, 68 Nordstrom-Robinson, 14, 154 bound: orbit, 132 Gilbert-Varshamov, 6, 149 perfect, 4 Goppa, 147 Preparata, 2, 173 linear programming, 6 quaternionic, 175 Singleton, 140 Reed-Muller, 7, 154 sphere packing, 4 Reed-Solomon, 14, 140, 214 Tsfasman-Vladut-Zink, 150 right closing sliding block, 127 right resolving sliding block, 127 capacity, 61, 134 self-dual, 3 channel capacity, 22 self-orthogonal, 3 channel impulse response, 21 sliding block, 96, 126 channel signal to noise function, 22 sliding-block-decodable finite-state, 129 charge-constrained, 44 spherical, 183 charge-RLL constraints, 44 trellis, 10, 31, 119 chordal , 180 Codemart, 180 class-4, 50 codewords, 2, 213 236 INDEX

coding gain: deterministic, 56 effective, 36 digital data storage, 41 nominal, 35 digital signature, 219 combined error-correction and modulation, digital sum variation (DSV), 44 85 discrete variables, 195 communication complexity, 217 discriminant, 27 compact, 112 distance-invariance, 157 complete, 112 dynamic programming, 51 complete out-splitting, 76 completion, 112 efficiency, 47 conjugacy, 96, 127 email, 185 constituent 1- or 2-dimensional constellation, encoder, 126: 24 block, 67 constraint length, 31 causal, 126 constraints (variables in experimental designs), complexity, 85 195 convolutional, 101, 131 continuous approximation, 24 deterministic, 68 continuous, shift-commuting map, 126 feedbackfree, 126 continuous variables, 195 finite-state, 42 controllability, 105 noncatastrophic, 47, 130 convenience stores, 181 polynomial convolutional, 132 correlated errors, 196 rate p : q block (S, a), 66 cover: rate p : q finite-state, 46 Fischer, 98, 124 state space encoder, 128 future, 98, 123 tagged (5, n), 65 Krieger, 98, 123 trellis section, 129 Shannon, 123 with finite-memory decoder, 129 Tom, 124 entropy, 100, 134 covering problem, 181 enumerative coding, 67 cycle in a graph, 54 error-locator function, 143 Euler's theorem, 187 dc-free, 44 evolution law, 118: decoder: deterministic, 121 state-dependent, 46 external behavior of, 119 decoding, 143: group,132 Berlekamp-Welch, 214 input/state/output (I/S/O), 129 Hermitian codes, 143 linear, 131 maximum-likelihood, 51 state space system of, 118 soft-decision decoding of block codes, 7 external behavior, 106, 117 Viterbi, 10, 51 density ratio, 49 factor map, 127 desarguesian plane, 166 factoring, 127 desarguesian spread, 165 factors through, 99 descendant edges, 75 finite automaton, 104, 118, 121: descendant states, 75 local, 123 design: minimal deterministic, 123 A-optimal, 194 non-ambiguous, 122 D-optimal, 194 finite memory, 56, 58 /-efficient, 194 finite-state coding theorem, 66 /-optimal, 194 finite-state, complete state behavior, 119 t - (v,fc,A), 20 finite-state inverse-to-coding theorem, 67 blocked, 197 finite-state transition diagram, 118 experimental, 179, 190 finite-type, 58 response-surface, 197 finitely equivalent, 101 sequential, 195 Franaszek algorithm, 63 spherical t-design, 181 free-concatenatable, 67 statistical, 182 frequency response, 21 design of experiments, 190 ftp, 186 designed distance, 147 future, 98 INDEX 237 , 3, 213 fundamental region, 26 Gosset, 190 fundamental volume, 27 graph, 97: Gosset (E8), 27 conjugacy-inducing, 123 integral, 28 DeBruijn, 134 mod 2, 29 directed, 97 unimodular, 28 edge, 76 Leech coordinate array, 33 higher 2-block, 76 left resolving, 98 higher edge, 134 Lennard-Jones potential, 185 higher power, 57, 134 line-set, 153 homomorphism, 98 linear trellis section, 131 irreducible, 59, 97, 121 look-ahead of sliding window decoder, 70 isomorphism, 54, 98 look-behind of sliding window decoder, 70 labeled, 53, 97, 118 labeled with no diamonds, 122 Mac Williams identity, 172 LFO labeled, 121 MAX-SNP, 211 lossless, 57 maximal volume, 181 lossless labeled, 122 measuring region, 196 lossless of finite order, 56 memory, 47 memory of, 56 messages, 213 primitive, 121 minimal energy, 182 right closing labeled, 121 minimal error probability, 181 right resolving conjugacy-inducing labeled, minimum distance, 3, 213 123 miracle octad generator, 15 right resolving labeled, 121 mixtures, 195 shift register, 134 modeling region, 194, 196 strongly regular, 169 models, 195 support, 62 monomial transformation, 4 Gray map, 171 multiple edges, 134 group trellis section, 132 netlib, 185 Hadamard matrices, 185 non-return-to-zero inverse (NRZI) precoding, , 2, 153 49 Hermitian curve, 141 nonambiguous, 104 hexacode, 14 nondeterministic polynomial time, 208 NP, 209 icosahedral symmetry, 185 NP-complete, 71, 209 imbedding, 128 NP-hard, 210 improved snub cube, 183 NP optimization function, 210 in-amalgamation, 100 NP search problem, 210 in-splitting, 76, 100 numerical integration, 181 induced vector, 78 instantaneously invertible trellis, 121 observability, 105 integrity, 217 optimality criteria, 196 interleaved NRZI (INRZI), 53 orbifold, 202 interpolation problem, 182 orthogonal arrays, 185 intra subset squared distance, 32 orthogonal geometry, 160 irreducible component, 60 orthogonal spread, 153 irreducible sink, 60 out-amalgamation, 100 irreducible source, 60 out-splitting, 75, 100 isometric embedding, 182, 200 isometry, 155, 160 P, 208 packing problem, 180 Jordan's theorem, 187 packings, 195 Kerdock set, 153 parallel-edgification, 103 kissing number, 28 parallel transitions, 134 parent edges, 75 language, 111 parent state, 75 lattice, 26: parity check matrix, 3 238 INDEX partial-response filtering, 50 security parameter, 217 path 7 in a graph, 54 self-testing/correcting pair, 220 pattern search, 185 sequence space, 115 PCP Theorem, 226 set partitioning, 32 peak detection, 49 Shannon capacity, 47 Perron-Probenius Theorem, 62 shell mapping, 31 place, 144 shift: point controllable state behavior, 120 edge sequence shift, 119 pollen, 182 edge shift, 97 polynomial space, 208 full shift, 96, 115 polynomial time, 208 graph shift, 119 power spectrum, 21 group shift, 102, 132 prediction variance, 193 homogeneous shift, 133 presentation, 97: L-step Shift of Finite Type (L-step SFT), C-minimal, 99 114 privacy, 217 shift-invariant, 112 PRML, 45 shift of finite type, 58, 98, 114 PRML constraints, 45 sofic shift, 54, 97, 119 proof/debate system: state sequence, 119 long, robust proof system, 226 topological Markov, 119 probabilistically checkable debate system, shift space, 96, 113: 229 irreducible, 116 probabilistically checkable proof systems, linear, 131 225 mixing, 116 transparent proof system, 226 topologically transitive, 116 program corrector, 219 signal constellation, 24 program tester, 219 signal power, 24 projective plane, 165 singular vector, 155 PSPACE, 208 skew-symmetric, 155 Slepian signal sets, 36 quadratic form, 155 small-bias probability spaces, 20 quadrature amplitude modulation, 23 snub cube, 183 quantization, 36 sphere, 179 quantizing problem, 181 spread, 165 quasi-equivalent, 162 spread set, 166 randomness as a resource, 214 state merging, 82 rate, 46 state sequence of a path, 54 ratio of peak to average power, 24 state-splitting algorithm, 42 realization, 106, 117: state behavior, 118 future induced canonical, 106 strongly complete, 115 group, 132 strongly point controllable state behavior, invertible, 123 121 linear, 131 symplectic space, 155 minimal, 106 symplectic spread, 153 past induced canonical, 106, 123 symbolic dynamics, 48 state, 106 syndrome, 143 reducible, 59 system: regular language, 104, 119 almost-finite type, 48 relative distance, 213 classical dynamical, 95 Riemann-Roch Theorem, 146 classical, 95 right closing, 56, 99 constrained, 42, 119 right resolving, 56, 98 controllable, 116 robust characterizations, 221 dynamical, 105, 111 round of out-splitting, 76 externally Induced state space, 123 runs, 196 finite memory, 114 finite type constraints, 48 Schwartz's Lemma, 213 free, 115 second moment, 183 group,132 INDEX 239 group state space, 132 strongly controllable, 116 input/output system, 125 strongly observable, 115 input/output system with output observ• symbolic dynamical, 96 able from input, 126 input-state-output, 106 Tammes problem, 182 input/state/output, 128 time-invariant, 112 irreducible constrained, 61 topological conjugacy, 127 L-complete, 114 totally singular subspace, 160 L-observable, 115 trace map, 157 linear, 131 transducer, 105, 129 linear input/state/output, 131 trellis, 118 memory less, 115 trellis diagram, 51, 118 non-anticipating input/output, 126 trellis section, 118 observable, 115 trellis that is invert ible with delay, 121 observable state space, 123 trim, 113 past-induced state space, 123 Turing machine, 208 presentation of, 54 Voronoi region, 26, 183 qth power of a labeled graph, 57 sofic, 54, 119 Waring's problem, 201 state space, 117 weight enumerator, 171 state space system of an evolution law, 118 Other Titles in This Series {Continued from the front of this publication)

20 J. P. LaSalle, editor, The influence of computing on mathematical research and education (University of Montana, August 1973) 19 J. T. Schwartz, editor, Mathematical aspects of (New York City, April 1966) 18 H. Grad, editor, Magneto-fluid and plasma dynamics (New York City, April 1965) 17 R. Finn, editor, Applications of nonlinear partial differential equations in (New York City, April 1964) 16 R. Bellman, editor, Stochastic processes in mathematical physics and engineering (New York City, April 1963) 15 N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962) 14 R. Bellman, editor, Mathematical problems in the biological sciences (New York City, April 1961) 13 R. Bellman, G. Birkhoff, and C. C. Lin, editors, Hydrodynamic instability (New York City, April 1960) 12 R. Jakobson, editor, Structure of language and its mathematical aspects (New York City, April 1960) 11 G. Birkhoff and E. P. Wigner, editors, Nuclear reactor theory (New York City, April 1959) 10 R. Bellman and M. Hall, Jr., editors, Combinatorial analysis (New York University, April 1957) 9 G. Birkhoff and R. E. Langer, editors, Orbit theory (Columbia University, April 1958) 8 L. M. Graves, editor, and its applications (University of Chicago, April 1956) 7 L. A. MacColl, editor, Applied probability (Polytechnic Institute of Brooklyn, April 1955) 6 J. H. Curtiss, editor, Numerical analysis (Santa Monica City College, August 1953) 5 A. E. Heins, editor, Wave motion and vibration theory (Carnegie Institute of Technology, June 1952) 4 M. H. Martin, editor, Fluid dynamics (University of Maryland, June 1951) 3 R. V. Churchill, editor, Elasticity (Uxiiversity of Michigan, June 1949) 2 A. H. Taub, editor, Electromagnetic theory (Massachusetts Institute of Technology, July 1948) 1 E. Reissner, editor, Non-linear problems in mechanics of continua (Brown University, August 1947)

ISBN 0-8218-0379-4