Graduate Texts in Mathematics Editorial Board F W Gehring C C Moore 86 P R Halmos (Managing Editor) J H van Lint Introduction to Coding Theory With Illustrations Springer Science+Business Media, LLC J H van Lint Eindhoven University of Technology Department of Mathematics Den Dolech 2, P.O Box 513 5600 MB Eindhoven The Netherlands Editorial Board P R Halmos F W Gehring c Managing Editor Indiana University Department of Mathematics Bloomington, IN 47401 U.S.A University of Michigan Department of Mathematics Ann Arbor, MI 48104 U.S.A University of California at Berkeley Department of Mathematics Berkeley, CA 94720 U.S.A C Moore AMS Classification (1980): 05-01, 68E99, 94A24 Library of Congress Cataloging in Publication Data Lint, Jacobus Hendricus van, 1932Introduction to coding theory (Graduate texts for mathematics; 86) Bibliography: p Includes index Coding theory Title II Series QA268.L57 519.4 82-3242 AACR2 © 1982 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc in 1982 Softcover reprint ofthe hardcover 1st edition 1982 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC 76 54 ISBN 978-3-662-08000-9 ISBN 978-3-662-07998-0 (eBook) DOI 10.1007/978-3-662-07998-0 Preface Coding theory is still a young subject One can safely say that it was born in 1948 It is not surprising that it has not yet become a fixed topic in the curriculum of most universities On the other hand, it is obvious that discrete mathematics is rapidly growing in importance The growing need for mathematicians and computer scientists in industry will lead to an increase in courses offered in the area of discrete mathematics One of the most suitable and fascinating is, indeed, coding theory So, it is not surprising that one more book on this subject now appears However, a little more justification and a little more history of the book are necessary A few years ago it was remarked at a meeting on coding theory that there was no book available which could be used for an introductory course on coding theory (mainly for mathematicians but also for students in engineering or computer science) The best known textbooks were either too old, too big, too technical, too much for specialists, etc The final remark was that my Springer Lecture Notes (# 201) were slightly obsolete and out of print Without realizing what I was getting into I announced that the statement was not true and proved this by showing several participants the book Inleiding in de Coderingstheorie, a little book based on the syllabus of a course given at the Mathematical Centre in Amsterdam in 1975 (M.C Syllabus 31) The course, which was a great success, was given by M R Best, A E Brouwer, P van Emde Boas, T M V Janssen, H W Lenstra Jr., A Schrijver, H C A van Tilborg and myself Since then the book has been used for a number of years at the Technological Universities of Delft and Eindhoven The comments above explain why it seemed reasonable (to me) to translate the Dutch book into English In the name of Springer-Verlag I thank the Mathematical Centre in Amsterdam for permission to so Of course it turned out to be more than a translation Much was rewritten or v Preface VI expanded, problems were changed and solutions were added, and a new chapter and several new proofs were included Nevertheless the M.e Syllabus (and the Springer Lecture Notes 201) are the basis of this book The book consists of three parts Chapter contains the prerequisite mathematical knowledge It is written in the style of a memory-refresher The reader who discovers topics which he does not know will get some idea about them but it is recommended that he also looks at standard textbooks on those topics Chapters to provide an introductory course in coding theory Finally, Chapters to 11 are introductions to special topics and can be used as supplementary reading or as a preparation for studying the literature Despite the youth of the subject, which is demonstrated by the fact that the papers mentioned in the references have 1974 as the average publication year, I have not considered it necessary to give credit to every author of the theorems, lemmas, etc Some have simply become standard knowledge It seems appropriate to mention a number of textbooks which I use regularly and which I would like to recommend to the student who would like to learn more than this introduction can offer First of all F J MacWilliams and N J A Sloane, The Theory of Error-Correcting Codes (reference [46]), which contains a much more extensive treatment of most of what is in this book and has 1500 references! For the more technically oriented student with an interest in decoding, complexity questions, etc E R Berlekamp's Algebraic Coding Theory (reference [2]) is a must For a very well-written mixture of information theory and coding theory I recommend: R J McEliece, The Theory of Information and Coding (reference [51]) In the present book very little attention is paid to the relation between coding theory and combinatorial mathematics For this the reader should consult P J Cameron and J H van Lint, Graphs, Codes and Designs (reference [11]) I sincerely hope that the time spent writing this book (instead of doing research) will be considered well invested Eindhoven July 1981 J H VAN LINT Contents CHAPTER I Mathematical Background 1.1 1.2 1.3 1.4 Algebra Krawtchouk Polynomials Combinatorial Theory Probability Theory 14 17 19 CHAPTER Shannon's Theorem 2.1 2.2 2.3 2.4 Introduction Shannon's Theorem Comments Problems 22 22 26 29 29 CHAPTER Linear Codes 31 3.1 3.2 3.3 3.4 3.5 3.6 3.7 31 33 36 37 38 40 40 Block Codes Linear Codes Hamming Codes Majority Logic Decoding Weight Enumerators Comments Problems vii viii Contents CHAPTER Some Good Codes 42 4.1 4.2 4.3 4.4 4.5 4.6 4.7 42 Hadamard Codes and Generalizations The Binary Golay Code The Ternary Golay Code Constructing Codes from Other Codes Reed-Muller Codes Comments Problems 43 44 45 47 52 52 CHAPTER Bounds on Codes 54 5.1 5.2 5.3 5.4 5.5 54 57 Introduction; The Gilbert Bound Upper Bounds The Linear Programming Bound Comments Problems 64 68 69 CHAPTER Cyclic Codes 70 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 70 72 73 74 77 Definitions Generator Matrix and Check Polynomial Zeros of a Cyclic Code The Idempotent of a Cyclic Code Other Representations of Cyclic Codes BCH Codes Decoding BCH Codes Reed-Solomon Codes Quadratic Residue Codes Comments Problems 80 84 85 86 89 89 CHAPTER Perfect Codes and Uniformly Packed Codes 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Lloyd's Theorem The Characteristic Polynomial of a Code Uniformly Packed Codes Examples of Uniformly Packed Codes Nonexistence Theorems Comments Problems 91 91 94 97 100 102 105 106 Contents IX CHAPTER Goppa Codes 107 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 107 108 109 110 111 Motivation Goppa Codes The Minimum Distance of Goppa Codes Asymptotic Behaviour of Goppa Codes Decoding Goppa Codes Generalized BCH Codes Comments Problems 112 114 115 CHAPTER Asymptotically Good Algebraic Codes 116 9.1 9.2 9.3 9.4 117 120 121 A Simple Nonconstructive Example Justesen Codes Comments Problems 116 CHAPTER 10 Arithmetic Codes 122 10.1 10.2 10.3 10.4 10.5 122 124 128 129 129 AN Codes The Arithmetic and Modular Weight Mandelbaum-Barrows Codes Comments Problems CHAPTER 11 Convolutional Codes 130 11.1 11.2 11.3 11.4 11.5 11.6 11.7 130 134 136 Introduction Decoding of Convolutional Codes An Analog of the Gilbert Bound for Some Convolutional Codes Construction of Convolutional Codes from Cyclic Block Codes Automorphisms of Convolutional Codes Comments Problems 137 140 142 143 Hints and Solutions to Problems 144 References 163 Index 167 Chapter Mathematical Background In order to be able to read this book a fairly thorough mathematical background is necessary In different chapters many different areas of mathematics playa role The most important one is certainly algebra but the reader must also know some facts from elementary number theory, probability theory and a number of concepts from combinatorial theory such as designs and geometries In the following sections we shall give a brief survey of the prerequisite knowledge Usually proofs will be omitted For these we refer to standard textbooks In some of the chapters we need a large number of facts concerning a not too well-known class of orthogonal polynomials, called Krawtchouk polynomials These properties are treated in Section 1.2 The notations which we use are fairly standard We mention a few which may not be generally known If C is a finite set we denote the number of elements of C by IC I If the expression B is the definition of concept A then we write A := B We use "iff" for "if and only if" An identity matrix is denoted by I and the matrix with all entries equal to one is J Similarly we abbreviate the vector with all coordinates (resp 1) by (resp 1) Instead of using [x] we write LxJ :=max{nEZln:::;; x} and use the symbol rxl for rounding upwards §1.1 Algebra We need only very little from elementary number theory We assume known that in N every number can be written in exactly one way as a product of prime numbers (if we ignore the order of the factors) If a divides b then we write alb If p is a prime number and prla but pr+l"ra then we write prlla.1f 158 Hints and Solutions to Problems Since on the left-hand side we have a term 2, the other term is not divisible by Therefore a = or a = If a = we find 2b(2 b + 1) + = 2c, i.e b = and n = corresponding to the code C = {(OO)} If a = we find 22b + 3· 2b + = 2c, i.e b = and then n = corresponding to the repetition code C = {(OOOOO), (l1111)} 7.7.6 First use Theorem 7.3.5 and (1.2.7) We find the equation 4X2 - 4(n + l)x + (n + n + 12) = 0, F=l1) with zeros X ,2 = t(n + ± It follows that n - 11 = m2 for some integer m From (7.3.6) we find 12·2n = ICI·(n + n + 12) = ICI(n + + m)(n + - m) So n + + m = a 2a + 1, n + - m = b 2P+ with ab = or First a = b = We find n + = 2a + 2P, m = 2a - 2P(ct > f3) and hence 2a + 2P - 12 = 22a _ 2a+P+ _ 22P, try I.e an obvious contradiction Next, try b = This leads to n + = a· 2a + 3· 2P, m = a· 2a - 3· 2P and hence 3· 2P(3 2P - 2a+1 - 1) + 2a(2 a - 1) + 12 = O If ct > then we must have f3 = and it follows that ct = Since ct ::; does not give a solution and the final case a = also does not yield anything we have proved that n + = 24 + 3.2 , i.e n = 27 The construction of such a code is similar to (7.4.2) Replace the form used in (7.4.2) by XIX2 + X3X4 + X5X6 + X5 + X6 = O The rest of the argument is the same We find a two-weight code of length 27 with weights 12 and 16 and then apply Theorem 7.3.7 Chapter 8.8.1 In Theorem 8.3.1 it was shown that if we replace g(z) by g(z) := Z + we get the same code So r(L, g) has dimension at least and minimum distance d ~ As was shown in the first part of Section 8.3, d might be larger We construct the parity check matrix H = (hoh ••• h7) where hi runs through the values (ct j + 1)-1 with (j, 15) = We find that H consists of all column vectors with a in the last position, i.e r(L, g) is the [8,4,4] extended Hamming code 8.8.2 Let a be a word of even weight in C By (6.5.2) the corresponding Mattson-Solomon polynomial A(X) is divisible by X By Theorem 8.6.1 the polynomial X n - A(X), i.e X- A(X), is divisible by g(X) Since C is cyclic 159 Hints and Solutions to Problems we find from (6.5.2) that X-I A(X) is also divisible by g(rxiX) for < i ~ n - If g(X) had a zero different from in any extension field of IF2 we would have n - distinct zeros of X-I A(X), a contradiction since X-I A(X) has degree excludes nine values of rx(x) and d > excludes the remaining 54 values of rx(x) This shows that for small n the construction is not so good! By (3.7.14) there is a [12, 7] extended lexicographically least code with d = In (4.7.3) we saw that a nonlinear code with n = 12, d = exists with even more words 9.4.2 Let rx(x) be a polynomial of weight Then in (a(x), rx(x)a(x» the weights of the two halves have the same parity So d < is only possible if there is a choice a(x) = Xi + xi such that rx(x)a(x) == mod(x - 1) This is 160 Hints and Solutions to Problems so if O(x) is periodic, i.e + x we have d = + X4 or x + x + x • For all other choices 9.4.3 Let the rate R satisfy 1/(1 + 1) < R ~ 1// (I EN) Let s be the least integer such that m/[(l + l)m - s] R We construct a code C by picking an I-tuple (0(1,0(2, ,0(/) E (1F mY and then forming (a, O(la, , O(/a) for all a E lFi and finally deleting the last s symbols The word length is n = (I + l)m - s A nonzero word C E C corresponds to 25 possible values of the I-tuple (O(t> ,0(/) To ensure a minimum distance An we must exclude ~ 25 Lid" CD values of (0(1' ,0(/) We are satisfied if this leaves us a choice for (0( 1, , 0(/), i.e if From Theorem 1.4.5 we find s + nH(A) < ml, i.e ml- s m H(A) < - - = - - = - R n n + 0(1), (m ~ co) Chapter 10 10.5.1 Consider the sequence r, r2, r3, There must be two elements in the sequence which are congruent mod A, say r" - rm == (mod A)(n > m) 10.5.2 Let m = r" - = AB, where A is a prime >r2 Suppose that r generates a subgroup H of IF~ with IH I = n which has {± c Ic = 1,2, , r - 1} as a complete set of coset representatives Consider the cyclic AN code C oflength n and base r Clearly every integer in the interval [1, m] has modular distance or to exactly one codeword So C is a perfect code (Since wm(A) we must have A > r2.) A trivial example for r = is the cyclic code {13, 26} Here we have taken m = 33 - and A = 13 The subgroup generated by in IF! has index and the coset representatives are ± and ± 10.5.3 We have 455 = Lf=o b)i where (b o, bt> , b ) = (2,1,2,1,2,1) The algorithm described in (10.2.3) replaces the initial 2, by -1, In this way we find the following sequence ofrepresentations: ° ° (2, 1,2, 1,2, 1) ~ ( -1,2,2, 1,2, 1) ~ (-1, -1,0,2, 1, 1) ~ (-1, -1,0, -1,0, 2) ~ (0, -1,0, -1,0, -1) So the representation in CNAF is 455 == - 273 = - - 33 - 35 161 Hints and Solutions to Problems 10.5.4 We check the conditions of Theorem 10.3.2 In IFf the element generates the subgroup {I, 3, 9, 5, 4}; multiplication by -1 yields the other five elements r" = 35 = 243 = + 11.22 So we have A = 22, in ternary representation A = + 1.3 + 2.3 • The CNAF of 22 is - 2.3 + 0.3 + 1.3 + 0.3 (mod 242) The code consists of ten words namely the cyclic shifts of (1, - 2,0, 1,0) resp ( -1,2,0, -1,0) All weights are Chapter 11 11.7.1 Using the notation of Section 11.1 we have + x + x + X4 The information stream 1 1 1··· would give Io(x) = (1 + X)-l, and G(x) = (1 + (X )2) + x(1 + x ) = hence T(x) = (1 + X2)-lG(X) = + x + x2, i.e the receiver would get 1 0 0··· Three errors in the initial positions would produce the zero signal and lead to infinitely many decoding errors 11.7.2 In Theorem 11.4.2 it is shown how this situation can arise Let h(x) = X4 + X + and g(x)h(x) = X l5 - We know that g(x) generates an irreducible cyclic code with minimum distance Consider the information sequence 1 0 0 0···, i.e Io(x) = h(x) Then we find T(x) = h(x )g(x) = (X l5 - l)h(x), which has weight By Theorem 11.4.2 this is the free distance In this example we have g(x) = Xll + x + X + x + x + x + X + Therefore Go(x) = + x + X4, Gl(x) = + x + x + x + x • The encoder is Figure 162 Hints and Solutions to Problems and 10 T = = 1 0 0 0··· yields as output 11 00 10 00 00 00 00 01 10 01 00 00 11.7.3 Consider any nonzero output sequence Clearly the initial 7-tuple is a nonzero codeword in the cyclic code generated by m3' Therefore it has weight ~ Similarly the final nonzero 7-tuple of output is a codeword in the code generated by mo ml and it is again not the zero word Hence it has weight ~4 Now suppose we have as input at t = the 4-tuple (1100) and subsequently only zeros The output is (1100)G, i.e (mlm3 + mOm3) + (m5ml)x, From the first two rows of Gin (11.5.6) we see that this is 0001011 followed by 1001011 and then zeros So the free distance is References Baumert, L D and McEliece, R J.: A Golay-Viterbi Concatenated Coding Scheme for MJS '77 JPL Technical Report 32-1526, pp 76-83 Pasadena, Calif.: Jet Propulsion Laboratory, 1973 Berlekamp, E R.: Algebraic Coding Theory New York: McGraw-Hill, 1968 Berlekamp, E R.: Decoding the Golay Code JPL Technical Report 32-1256, Vol IX, pp 81-85 Pasadena, Calif.: Jet Propulsion Laboratory, 1972 Berlekamp, E R.: Goppa codes IEEE Trans Info Theory, 19, pp 590-592 (1973) Berlekamp, E R and Moreno, 0.: Extended double-error-correcting binary Goppa codes are cyclic IEEE Trans Info Theory, 19, pp 817-818 (1973) Best, M R., Brouwer, A E., MacWilliams, F J., Odlyzko, A M and Sloane, N J A.: Bounds for binary codes oflength less than 25 IEEE Trans Info Theory, 23, pp 81-93 (1977) Best, M R.: On the Existence of Perfect Codes Report ZN 82/78 Amsterdam: Mathematical Centre, 1978 Best, M R.: Binary codes with a minimum distance of four IEEE Trans Info Theory, 26, pp 738-742 (1980) Bussey, W H.: Galois field tables for p ~ 169 Bull Amer Math Soc., 12, pp 22-38 (1905) lO Bussey, W H.: Tables of Galois fields of order less than 1,000 Bull Amer Math Soc., 16, pp 188-206 (19lO) 11 Cameron, P J and van Lint, J H.: Graphs, Codes and Designs London Math Soc Lecture Note Series, Vol 43 Cambridge: Cambridge Univ Press, 1980 12 Chen, C L., Chien, R T and Liu, C K.: On the binary representation form of certain integers SIAM J Appl Math., 26, pp 285-293 (1974) 13 Chien, R T and Choy, D M.: Algebraic generalization of BCH-Goppa-Helgert codes IEEE Trans Info Theory, 21, pp 70-79 (1975) 14 Clark, W E and Liang, J J.: On arithmetic weight for a general radix representation of integers IEEE Trans Info Theory, 19, pp 823-826 (1973) 15 Clark, W E and Liang, J J.: On modular weight and cyclic nonadjacent forms for arithmetic codes IEEE Trans Info Theory, 20, pp 767-770 (1974) 16 Curtis, C W and Reiner, I : Representation Theory ofFinite Groups and Associative Algebras New York-London: Interscience, 1962 163 164 References 17 Cvetkovic, D M and van Lint, H.: An elementary proof of Lloyd's theorem Proc Kon Ned Akad v Wetensch (A), 80, pp 6-10 (1977) 18 Delsarte, P.: An algebraic approach to coding theory Philips Research Reports Supplements, 10 (1973) 19 Delasarte, P and Goethals, J.-M.: Unrestricted codes with the Golay parameters are unique Discrete Math., 12, pp 211-224 (1975) 20 Elias, P.: Coding for Noisy Channels IRE Conv Record, part 4, pp 37-46 21 Feller, W.: An Introduction to Probability Theory and Its Applications, Vol I New York-London: Wiley, 1950 22 Forney, G D.: Concatenated Codes Cambridge, Mass.: MIT Press, 1966 23 Forney, G D.: Convolutional codes I: algebraic structure IEEE Trans Info Theory, 16, pp 720-738 (1970); Ibid., 17, 360 (1971) 24 Gallagher, R G.: Information Theory and Reliable Communication New York: Wiley, 1968 25 Goethals, J.-M and van Tilborg, H C A.: Uniformly packed codes Philips Research Reports, 30, pp 9-36 (1975) 26 Goethals, J.-M.: The extended Nadler code is unique IEEE Trans Info Theory, 23, pp 132-135 (1977) 27 Goppa, V D.: A new class of linear error-correcting codes Problems of Info Transmission, 6, pp 207-212 (1970) 28 Goto, M.: A note on perfect decimal AN codes Info and Control, 29, pp 385-387 (1975) 29 Goto, M and Fukumara, T.: Perfect nonbinary AN codes with distance three Info and Control, 27, pp 336-348 (1975) 30 Graham, R L and Sloane, N J A.: Lower bounds for constant weight codes IEEE Trans Info Theory, 26, pp 37-40 (1980) 31 Gritsenko, V M.: Nonbinary arithmetic correcting codes Problems of Info Transmission, 5, pp 15-22 (1969) 32 Hall, M.: Combinatorial Theory New York-London-Sydney-Toronto: Wiley (second printing), 1980 33 Hartmann, C R P and Tzeng, K K.: Generalizations of the BCH bound Info and Control, 20, pp 489-498 (1972) 34 He1gert, H J and Stinaff, R D.: Minimum distance bounds for binary linear codes IEEE Trans Info Theory, 19, pp 344-356 (1973) 35 Helgert, H J.: AIternant codes Info and Control, 26, pp 369-380 (1974) 36 Jackson, D.: Fourier Series and Orthogonal Polynomials Carus Math Monographs, Vol Math Assoc of America, 1941 37 Justesen, J.: A class of constructive asymptotically good algebraic codes IEEE Trans Info Theory, 18, pp 652-656 (1972) 38 Justesen, J.: An algebraic construction of rate I/v convolutional codes IEEE Trans Info Theory, 21, 577-580 (1975) 39 Kasami, T.: An upper bound on kin for affine invariant codes with fixed din IEEE Trans Info Theory, 15, pp 171-176 (1969) 40 Levenshtein, V I.: Minimum redundancy of binary error-correcting codes Info and Control, 28, pp 268-291 (1975) 41 van Lint, J H.: Nonexistence theorems for perfect error-correcting-codes In: Computers in Algebra and Number Theory, Vol IV (SIAMS-AMS Proceedings) 1971 42 van Lint, J H.: Coding Theory Springer Lecture Notes, Vol 201, Berlin-Heidelberg-New York: Springer, 1971 43 van Lint, J H.: A new description of the Nadler code IEEE Trans Info Theory, 18, pp 825-826 (1972) 44 van Lint, J H.: A survey of perfect codes Rocky Mountain J Math., 5, pp 199-224 (1975) 45 van Lint, J H and MacWilliams, F J.: Generalized quadratic residue codes IEEE Trans Info Theory, 24, pp 730-737 (1978) References 165 46 MacWilliams, F J and Sloane, N J A.: The Theory of Error-correcting Codes Amsterdam-New York-Oxford: North Holland, 1977 47 Massey, J L.: Threshold Decoding Cambridge, Mass.: MIT Press, 1963 48 Massey, J L and Garcia, O N : Error-correcting codes in computer arithmetic In: Advances in Information Systems Science, Vol 4, Ch (Edited by J T Ton) New York: Plenum Press, 1972 49 Massey, J L., Costello, D J and Justesen, J.: Polynomial weights and code construction IEEE Trans Info Theory, 19, pp 101-110 (1973) 50 McEliece, R J., Rodemich, E R., Rumsey, H C and Welch, L R.: New upper bounds on the rate of a code via the Deisarte-MacWilIiams inequalities IEEE Trans Info Theory, 23, pp 157-166 (1977) 51 McEliece, R J.: The Theory of I"formation and Coding Encyclopedia of Math and its Applications, Vol Reading, Mass.: Addison-Wesley, 1977 52 McEliece, R J.: The bounds of Delsarte and Lovasz and their applications to coding theory In: Algebraic Coding Theory and Applications (Edited by G Longo, CISM Courses and Lectures, Vol 258 Wien-New York: Springer, 1979 53 Peterson, W W and Weldon, E J.: Error-correcting Codes (2nd ed.) Cambridge, Mass.: MIT Press, 1972 54 Piret, Ph.: Structure and constructions of cyclic convolutional codes IEEE Trans Info Theory, 22, pp 147-155 (1976) 55 Piret, Ph.: Algebraic properties of convolutional codes with automorphisms Ph.D Dissertation Univ Catholique de Louvain, 1977 56 Posner, E C.: Combinatorial structures in planetary reconnaissance In: Error Correcting Codes (Edited by H B Mann pp 15-46 New York-London-SydneyToronto: Wiley, 1968 57 Preparata, F P.: A class of optimum nonlinear double-error-correcting codes Info and Control, 13, pp 378-400 (1968) 58 Rao, T R N.: Error Coding for Arithmetic Processors New York-London: Academic Press, 1974 59 Roos, c.: On the structure of convolutional and cyclic convolutional codes IEEE Trans Info Theory, 25, pp 676-683 (1979) 60 Schalkwijk, J P M., Vinck, A J and Post, K A.: Syndrome decoding of binary rate kin convolutional codes IEEE Trans Info Theory, 24, pp 553-562 (1978) 61 Selmer, E S.: Linear recurrence relations over finite fields Univ of Bergen, Norway: Dept of Math., 1966 62 Shannon, C E.: A mathematical theory of communication Bell Syst Tech J., 27, pp 379-423, 623-656 (1948) 63 Sidelnikov, V M.: Upper bounds for the number of points ofa binary code with a specified code distance Info and Control, 28, pp 292-303 (1975) 64 Sloane, N J A and Whitehead, D S.: A new family of single-error-correcting codes IEEE Trans Info Theory, 16, pp 717-719 (1970) 65 Sloane, N J A., Reddy, S M and Chen, C L.: New binary codes IEEE Trans Info Theory, 18, pp 503-510 (1972) 66 Solomon, G and van Tilborg, H C A.: A connection between block and convolutional codes SIAM J Appl Math., 37, pp 358-369 (1979) 67 Szego, G.: Orthogonal Polynomials Colloquium Publications, Vol 23 New York: Amer Math Soc (revised edition), 1959 68 Tietavliinen, A.: On the nonexistence of perfect codes over finite fields SIAM J Appl Math., 24, pp 88-96 (1973) 69 van Tilborg, H C A.: Uniformly packed codes Thesis, Eindhoven Univ of Technology, 1976 70 Tricomi, F G.: Vorlesungen uber Orthogonalreihen Grundlehren d math Wiss Band 76 Berlin-Heidelberg-New York: Springer, 1970 71 Tzeng, K K and Zimmerman, K P.: On extending Goppa codes to cyclic codes IEEE Trans Info Theory, 21, pp 712-716 (1975) Index lX(b), 55 adder (mod 2), 131 admissible pair, 125 affine geometry, 17 permutation group, subspace, 17 transformation, 17 AG(m, q), 17 AGL(1, q"'), 83 AGL(m, 2), 51 algebra, alphabet, 31 A(n, d), 54 A(n, d, w), 62 arithmetic distance, 122 weight, 122, 124 automorphism group, 51, 83,141 Baker, 101 Barrows, 128 basis, Bell Laboratories, 29 Berlekamp, 29, 85, 114 decoder, 85 Best, 46, 68, 102 binary entropy, 20 symmetric channel, 24 binomial distribution, 20 bit, 23 block design, 17 length, 31 Bose, 80 bound BCH,80 Carlitz-Uchiyama,89 Elias, 61 Gilbert- Varshamov, 54, 56, 111, 117, 136 Grey, 69 Griesmer,59 Hamming, 59 Johnson, 62 linear programming, 64 McEliece, 64 Plotkin, 57 Singleton, 57 sphere packing, 59 burst, 86 Cameron, 52 CCC,141 character, 13 principal, 14 characteristic numbers, 96 polynomial, 94, 96 Chebyshev's inequality, 19 check polynomial, 72 Chen, 129 167 168 Chien, 113, 129 Choy,l13 Christoffel-Darboux formula, 16 Clark,129 CNAF,I27 code, 32 alternant, 114 AN,122 arithmetic, 122 asymptotically good, 116 BCH, 80,108 narrow sense, 80 primitive, 80 Best, 46 block,31 catastrophic, 133 completely regular, 75 concatenated, 117 constacyclic, 70 convolutional, 31, 130, 134 cyclic, 70 AN,124 convolutional, 141 direct product, 41 double circulant, 121 dual, 34 equidistant, 58, 128 equivalent, 33 error-correcting, 22 error -detecting, 23 extended, 36, 45 generalized BCH, 113 Golay binary, 43, 89 ternary, 44, 52, 53 Goppa,108 group, 33 Hadamard, 42, 52, 100, 116 Hamming, 36, 50, 73 inner, 117 irreducible cyclic, 7I Justesen, 117 lexicographically least, 41 linear, 33 Mandelbaum-Barrows, 128 maximal, 54 cyclic, 71 maximum distance separable, 54, 146 MDS, 54, 57 minimal cyclic, 71, 129 modular arithmetic, 123 Nadler 45 narrow sense BCH, 80 nearly perfect, 98 Index negacyclic, 70 Nordstrom-Robinson, 45, 102, 106 optimal, 54 outer, 117 perfect, 32, 36,91, 124, 129 Preparata, 101 primitive BCH, 80 projective, 36 punctured, 45 QR,86 quadratic residue, 86 quasi-perfect, 35 Reed-Muller, 47, 49 Reed-Solomon, 85, 117 regular, 95 repetition, 24 residual, 46 RM,47 self-dual, 34 separable, 34 shortened, 45 Srivastava, 114 symmetry, 150 systematic, 33 ternary, 32 trivial, 32 two-weight, 99 uniformly packed, 97 uniquely decodable, 41 codeword, 23 conference matrix, 18 constraint length, 133 coset, leader, 35 representative, covering radius, 32 Cvetkovic, 91 cyclic nonadjacent form, 127 cyclotomic coset, 75 decision (hard, soft), 24 decoder, 23 Berlekamp, 85, III decoding BCH codes, 84 complete, 23 Goppa codes, III incomplete, 23 majority logic, 37 maximum likelihood, 26 multistep majority, 52 RM codes, 51 Viterbi, 134 169 Index Delsarte, 64, 106 derivative, II design block,17 to, 17,43,150 direct product, 41 distance, 31 arithmetic, 122 distribution, 65 enumerator, 94 external, 97 free, 133 Hamming, 31 minimum, 32 Elias, 60, 142 encoder, 23 entropy, 20, 55 erasure, 24 error, 23 locator polynomial, 84, III Euler indicator, expected value, 19 external distance, 97 factor group, Feller, 19 field,4 finite field, 4, fiat, 17 Forney, 117 Fukumara, 129 Garcia, 129 generator of a cyclic group of AN code, 124 matrix, 33 of a cyclic code, 72 polynomial of a convolutional code, 132 of a cyclic code, 72 Gilbert, 29 Goethals, 45, 98 Golay, 52 Goppa, 114 polynomial, 108 Goto, 129 Griesmer, 129 Gritsenko, 129 group, abelian, algebra, 5, 95 commutative, cyclic, transitive, Hadamard matrix, 18 Hall, 17, 18 Hamming, 29, 36 Hartmann, 81 Helgert, 46, 114 Hocquenghem, 80 Hq(x),55 hyperplane, 17 ideal,3 principal, idempotent, 74 of a QR code, 88 incidence matrix, 17 independent variables, 19 vectors, information rate, 26, 32 symbol,33 inner distribution, 65 product, irreducible polynomial, Jackson, 14 Jet Propulsion Laboratory, 22 Justesen, 1l7, 139 Kasami,11O Krawtchouk expansion, 16 polynomial, 14,64,92 Kronecker product, 18 Lenstra, 129 Levenshtein, 68 Liang, 129 linear programming bound, 64 recurring sequence, 79 170 Lint, van, 52, 89, 91, 102, 150 Liu,129 Lloyd, 29, 91 theorem, 91, 98 MacWilliams, 29, 39, 89 Mandelbaum, 128 Mariner, 22 Massey, 40, 47, 129 Mattson-Solomon polynomial, 78 McEliece, 67, 68 mean, 19 memory, 131, 133 minimal polynomial, 10 modular distance, 123 weight, 123, 124 Moebius function, inversion formula, monic polynomial, Moreno, 114 Muller, 47 multiplicative group of a field, NAF,125 nonadjacent form, 125 oreler, orthogonal parity checks, 37 outer distribution, 94 Paley, 18 matrix, 18,42 parity check equation, 34 matrix, 34 symbol, 33 permutation, matrix, Peterson, 129 PG(n, q), 17 Piret, 140 Pless, 150 polynomials, II Post, 143 Preparata, 10 I primitive element, idempotent, 75 polynomial, 10 root of unity, Index principal character, 14 ideal,6 ideal ring, projective geometry, 17 plane, 17 PSL(2, n), 88 quadratic residue, 12 Rao, 129 Ray-Chaudhuri, 80 redundancy, 22 Reed, 47 representative, residue class, ring, ring, Rodemich, 67 Roos, 81, 140 iJt(r, m), 49 Rumsey, 67 Schalkwijk, 143 Shannon, 24, 29 theorem, 22, 26 shift register, 131 Sidelnikov, 68 Slepian, 29, 40 Sloane, 29 Snover, 43 Solomon, 143 sphere, 27, 32 packing condition, 32 Srivastava, 114 standard deviation, 19 form, 33 state diagram, 131 Steiner system, 17 Stinaff,46 Stirling's formula, 20 subspace, symbol error probability, 29 symmetric group, syndrome, 34, III Szego, 14 171 Index Tietiiviiinen, 102 Tilborg, van, 98, 102, 143 trace, 13, 77 Tricomi,14 Tzeng, 81, 114 Vandermonde determinant, 79 variance, 19 Varshamov,56 vector space, Vinck,143 Viterbi algorithm, 135 Vq(n, r), 55 weight, 32 distribution, 38 enumerator, 38 Weil,89 Welch, 67 Weldon, 129 Wilson, 101 word, 23 length, 31 zero of a cyclic code, 73 Zimmerman, 114 Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 TAKEun/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra 3rd printing MACLANE Categories for the Working Mathematician HUGHEs/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 3rd printing revised COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules GOLUBITSKy/GuILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNEs/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis 5th printing MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol I ZARISKUSAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I: Basic Concepts JACOBSON Lectures in Abstract Algebra II: Linear Algebra JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENy/SNELUKNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE Probability Theory I 4th ed LOEVE Probability Theory 11 4th ed 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 MOISE Geometric Topology in Dimensions and SACHs/WU General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem 2nd printing KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRA vERlW ATKIN S Combinatorics with Emphasis on the Theory of Graphs BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis MASSEY Algebraic Topology: An Introduction CROWELL/Fox Introduction to Knot Theory KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions LANG Cyclotomic Fields ARNOLD Mathematical Methods in Classical Mechanics 3rd printing WHITEHEAD Elements of Homotopy Theory KARGAPOLOV/MERZLJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory FARKAsiKRA Riemann Surfaces STILLWELL Classical Topology and Combinatorial Group Theory HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 2nd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras IITAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRlsiSANKAPPANA VAR A Course in Universal Algebra WAL TERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups FORSTER Lectures on Riemann Surfaces BOTT/Tu Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields IRELAND/RoSEN A Classical Introduction Modern Number Theory EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory BROWN Cohomology of Groups PIERCE Associative Algebras ... Algebra Krawtchouk Polynomials Combinatorial Theory Probability Theory 14 17 19 CHAPTER Shannon's Theorem 2. 1 2. 2 2. 3 2. 4 Introduction Shannon's Theorem Comments Problems 22 22 26 29 29 CHAPTER Linear... the word is changed into something which resembles the correct word more 22 23 ? ?2. 1 Introduction than it resembles any other word we know This is the essence of the theory to be treated in this... Mathematics Editorial Board F W Gehring C C Moore 86 P R Halmos (Managing Editor) J H van Lint Introduction to Coding Theory With Illustrations Springer Science+Business Media, LLC J H van Lint