实例介绍
【实例简介】
This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings,
C North-Holland Publishing Company 1977 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner L CC Number: 76-41296 ISBN:0444850090 and0444850104 Published by North-Holland Publishing Company Amsterdan· New york· Oxford Sole distributors for the U.S.A. and Canada Elsevier/North-Holland Inc 52 Vanderbilt Avenue New York. NY 10017 Library of Congress Cataloging in Publication Data Mac Williams, Florence Jessie. 1917 The theory of error-correcting codes 1. Error-correcting codes (Information theory) Sloane, Neil James Alexander, 1939- joint author IL. Title QA268.M3 5194 76-41296 ISBN0444850090 and0444850104 Second printing 1978 Third printing 1981 PRINTED IN THE NETHERLANDS P reface Coding theory began in the late 1940s with the work of Golay, Hamming and Shannon. Although it has its origins in an engineering problem, the subject has developed by using more and more sophisticated mathematical techniques. It is our goal to present the theory of error-correcting codes in a simple, easily understandable manner, and yet also to cover all the important aspects of the subject. Thus the reader will find both the simpler families of codes-for example, Hamming, BCH, cyclic and Reed-Muller codes-dis- cussed in some detail, together with encoding and decoding methods, as well as more advanced topics such as quadratic residue, Golay, Goppa, alternant, Kerdock, Preparata and self-dual codes and association schemes Our treatment of bounds on the size of a code is similarly thorough We discuss both the simpler results-the sphere-packing, Plotkin, Elias and Garshamov bounds-as well as the very powerful linear programming method and the McEliece -Rodemich-Rumsey-Welch bound, the best asymptotic result known. An appendix gives tables of bounds and of the best codes presently known of length up to 512 Having two authors has helped to keep things simple: by the time we both understand a chapter, it is usually transparent. Therefore this book can be used both by the beginner and by the expert, as an introductory textbook and as a reference book, and both by the engineer and the mathematician. of course this has not resulted in a thin book and so we suggest the following meilun An elementary first course on coding theory for mathematicians: Ch 1, Ch 2(56 up to Theorem 22), Ch 3, Ch 4(381-5), Ch 5(to Problem 5), Ch 7(not §87,8),Ch.8(§§1-3),Ch.9(§81,4),Ch.12(§8),Ch.13(§§1-3),Ch.14 (§§1-3) A second course for mathematicians:Ch.2(§§1-6,8),Ch.4(§6,7and art of 8), Ch 5(to Problem 6, and $83, 4, 5, 7), Ch 6(S$1-3, 10, omitting the Preface proof of Theorem33),Ch.8(§5,6),Ch.9(§§2,3,5),Ch.10(§§1-5,11),Ch.11, Ch. 13(584, 5,9),Ch. 16(891-6), Ch 17(87, up to Theorem 35), Ch. 19(SS1 An elementary first course on coding theory for engineers: Ch. 1, Ch.3, Ch.4(§§1-5),Ch.5( to problem5),Ch.7(not§7),Ch.9(§51,4,6),Ch.10 (§§1,2,5,6,7,10),Ch.13(§§1-3,6,7),Ch.14(§§1,2,4) A second course for engineers: Ch. 2(3$1-6), Ch. 8(5$1-3, 5, 6), Ch. 9 (S§2,3,5),Ch.10(§11),Ch.12(s1-3,8,9),Ch.l6(§§1,2,4,6,9),Ch.17(7, up to Theorem 35). There is then a lot of rich food left for an advanced course the rest of Chapters 2, 6, 1 l and 14, followed by Chapters 15, 18, 19, 20 and 21-a feast The following are the principal codes discussed Alternant Ch, 12 BCH,Ch.3,§S1,3;Ch.7,§6;Ch.8,§5;Ch.9;Ch.21,§8 Chien-Choy generalized BCH, Ch. 12, $7 Concatenated,Ch.10,§1l;Ch.18,§§5,8; Conference matrix, Ch 2, 84 Cyclic, Ch. 7, Ch. 8 Delsarte-Goethals, Ch. 15, $5 Difference-set cyclic, Ch. 13, $8 Double circulant and quasi-cyclic, Ch 16, $86-8 Euclidean and projective geometry, Ch. 13, $8 Goethals generalized Preparata, Ch. 15, 87 Golay(binary), Ch. 2, 86; Ch. 16, 82; Ch. 20 Golay (ternary), Ch. 16, 82: Ch. 20 Goppa,Ch.12,§83-5 Hadamard, Ch. 2. $3 Hamming,Ch.1,§7,Ch.7,§3 and Problem8; Irreducible or minimal cyclic,Ch.8,§§3,4; Justesen,Ch.10,§ll; Kerdock,Ch.2,§8;Ch.15,§5; Maximal distance separable, Ch. I1 Nordstrom- Robinson,Ch.2,§8;Ch.15,§§5,6 Pless symmetry, Ch. 16,88 Preparata,Ch2,§8;Ch.15,$6;Ch.18,§7.3; Product,Ch.18,§§2-6; Quadratic residue, Ch. 16 Redundant residue, Ch. 10,89 Reed-Muller, Ch. 1, $9 Chs. 13-15 Reed-Solomon, Ch. 10 Preface v Self-dual, Ch. 19 Single-error-correcting nonlinear, Ch. 2, 87; Ch. 18, 87.3 Srivastava, ch. 12, 86 Encoding methods are given for Linear codes, ch.1,82: Cyclic codes, Ch.7, 88 Reed-Solomon codes, Ch. 10, $7 Reed- Muller codes,Ch.13,§§6,7;Ch.14,§4. Decoding methods are given for Linear codes, Ch. 1, s83,4 Hamming codes, Ch. 1, 87 BCH codes, Ch3, 83; Ch9, $6: Ch. 12, 59; Reed-Solomon codes, Ch.10.$10 Alternant (including BCH, Goppa, Srivastava and Chien-Choy generalized BCH codes)Ch.12,§9 Quadratic residue codes, ch. 16, 89; Cyclic codes, Ch. 16, 89 while other decoding methods are mentioned in the notes to ch. 16 When reading the book, keep in mind this piece of advice, which should be given in every preface: if you get stuck on a section, skip it, but keep reading! Dont hesitate to skip the proof of a theorem: we often do. Starred sections are difficult or dull, and can be omitted on the first (or even second) reading. The book ends with an extensive bibliography. Because coding theory overlaps with so many other subjects (computers, digital systems, group theory, number theory, the design of experiments, etc. )relevant papers may be found almost anywhere in the scientific literature. Unfortunately this means that the usual indexing and reviewing journals are not always helpful We have therefore felt an obligation to give a fairly comprehensive bi biography. The notes at the ends of the chapters give sources for the theorems, problems and tables, as well as small bibliographies for some of the topics covered (or not covered) in the chapter Only block codes for correcting random errors are discussed we say little about codes for correcting other kinds of errors(bursts or transpositions )or about variable length codes convolutional codes or source codes (see the Preface otes to Ch. 1). Furthermore we have often considered only binary codes which makes the theory a lot simpler. most writers take the opposite point of view: they think in binary but publish their results over arbitrary fields There are a few topics which were included in the original plan for the book but have been reluctantly omitted for reasons of space: Gray codes and snake -in-the- box codes -see adelson et al. [5, 6] Buchner [210], Cavior [2531, Chien et al. [2901, Cohn [2991, Danzer and Klee 328], Davies [335], Douglas [382, 383], Even [4131, Flores [432], Gardner [4681, Gilbert [4811, Guy [571], Harper [605], Klee [764-767l, Mecklenberg et al. [951], Mills [9561, Preparata and Nievergelt [1083], Singleton [1215], Tang and Liu [1307], Vasilev [1367], Wyner [1440] and Yuen [1448, 1449] (ii Comma-free codes-see Ball and Cummings [60, 61, Baumert and Cantor [85], Crick et al. [3161, Eastman [399, golomb [523, pp. 118-122], Golomb et al. [5281, Hall [587, pp. 11-12, Jiggs [692], Miyakawa and moriya [967, Niho [992] and Redinbo and Walcott [1102]. See also the remarks on codes for synchronizing in the Notes to Ch. 1 (il) Codes with unequal error protection -see Gore and Kilgus [549] Kilgus and Gore [761] and Mandelbaum [9011 (iv) Coding for channels with feed back -see Berlekamp [1241, Horstein 664] and Schalkwijk et al. [1153-1155] v) Codes for the Gaussian channel-see Biglieri et al. [148-151], Blake [155, 156, 1581, Blake and Mullin [162], Chadwick et al. [256, 257, Gallager [4641, Ingemarsson [6831, Landau [7911, Ottoson [1017], Shannon [11911, Slepian [1221-1223] and Zetterberg [1456 (vi) The complexity of decoding-see Bajoga and Walbesser [59], Chaitin [257a-258a1, Gelfand et al. [471], Groth [564], Justesen [7061, Kolmogorov [774a1, Marguinaud [9161, Martin-Lof [917a], Pinsker [1046a], Sarwate [11451 and Savage [1149-1152a vi) The connections between coding theory and the packing of equal spheres in n-dimensional euclidean space-see Leech [803-805],[807, Leech and Sloane [808-810 and Sloane [1226 The following books and monographs on coding theory are our predeces sors: Berlekamp [113, 116], Blake and Mullin [162], Cameron and van lint [234, Golomb [522], Lin [8341, Van Lint [848], Massey [922a], Peterson [1036a, Peterson and Weldon [10401, Solomon [1251] and sloane [1227a]; while the following collections contain some of the papers in the bibliography Berlekamp [126], Blake [157, the special issues [377a, 678, 679], Hartnett [620], Mann [909]and Slepian [1224. see also the bibliography [1022 We owe a considerable debt to several friends who read the first draft very carefully, made numerous corrections and improvements, and frequently saved us from dreadful blunders. In particular we should like to thank IF Blake, P. Delsarte, J -M. Goethals, R.L. Graham, J.H. van Lint, G. Longo C L. Mallows, J. McKay, V. Pless, H.O. Pollak, L D. Rudolph, D.W. Sar wate, many other colleagues at Bell Labs, and especially A m. odlyzko for Preface their help. Not all of their suggestions have been followed, however, and the authors are fully responsible for the remaining errors. ( This conventional remark is to be taken seriously. ) We should also like to thank all the typists at Bell Labs who have helped with the book at various times, our secretary Peggy van Ness who has helped in countless ways, and above all Marion Messersmith who has typed and retyped most of the chapters Sam lomonaco has very kindly helped us check the galley proofs Preface to the third printing We should like to thank many friends who have pointed out errors and misprints. The corrections have either been made in the text or are listed below A Russian edition was published in 1979 by Svyaz(Moscow), and we are extremely grateful to L. A. Bassalygo, I. T. Grushko and v. a. Zinov'ev for producing a very careful translation. They supplied us with an extensive list of corrections. They also point out (in footnotes to the russian edition a number of places we did not cite the earliest source for a theorem we have corrected the most glaring omissions, but future historians of coding theory should also consult the russian edition Problem 17, page 75. Shmuel Schreiber has pointed out that not all ways of choosing the matrices A, B, C, d work. one choice which does work is 0001 0010 0100 1000 A=/1000 0100 0010 0010 000 1000 D=/007 0100 0100 1000 0001」 010 Page 36, notes to 82. Add after Wu [1435, 1436: K. sh. Zigangirov, Some sequential decoding procedures, Problems of Information Transmission, 2(4) (1966)1-10; and K Sh Zigangirov, Sequential Decoding Procedures(Svyaz, Moscow, 1974) Page 72, Research problem 2. 4. It is now known that A(10, 4)=40,72s A(11, 4)<79, and 144= A(12, 4)s158. See M.R. Best, Binary codes with a minimum distance of four, IEEE Trans. Info. Theory, Vol. IT- 26(6)(November 1980),738-743 Page 123, Research problem 4.1. Self-contained proofs have been given by O Moreno, On primitive elements of Trace 1 in GF(2m), Discrete Math., to appear and L. R. vermani, Primitive elements with nonzero trace, preprint Preface to the third printing Chapter 6, pp. 156 and 180. Theorem 33 was proved independently by Zinov'ev and Leont'ev [1472] Page 166, Research problem 6. I has been settled in the affirmative by I. I Dumer, A remark about codes and tactical configurations, Math. notes, to appear;and by C Roos, Some results on t-constant codes, IEEE Trans. Info Theory, to appear Page 175. Research problem 6. 3 has also been solved by dumer and roos [op cit Page 178-179 Research problems 6.4 and 6.5 have been solved by Dumer [op cit]. The answer to 6.5 is No Page 267, Fig9.1.R.E. Kibler(private communication) has pointed out that the asterisks may be removed from the entries [127, 29, 43], [255, 45, 87]and [255, 37,91] since there are minimum weight codewords which are low-degree multiples of the generator polynomial Page 280, Research problem 9.4. T Helleseth(IEEE Trans. Info. Theory, Vol IT.25( 1979)361-362)has shown that no other binary primitive bCh codes are quasi-perfect Page 299, Research problem 10.1. R.E. Kibler(private communication)has found a large number of such codes Page 323. The proof of Theorem 9 is valid only for g= 2m. For odd g the code need not be cyclic. See G. Falkner, w. Heise, B. Kowol and E Zehender, on the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Universita di modena, to appear. Page 394, line 9 from the bottom. As pointed out by Massey in[918, P. 100], the number of majority gates required in an L-step decoder need never exceed the dimension of the code Page 479. Research problem 15.2 is also solved in I. I. Dumer, Some new uniformly packed codes, in: Proceedings MFTI, Radiotechnology and elec- tronics Series(MFTI, Moscow, 1976)pp. 72-78 Page 546. The answer to Research problem 17.6 is No, and in fact R. E Kibler Some new constant weight codes, IEEE Trans. Info. Theory, Vol. IT-26(May 1980)364-365, hows that27≤A(24,l0,8)≤68 Appendix a, Figures I and 3. For later versions of the tables of A(n, d)and A(n, d, w)see R. L. Graham and n.j. A Sloane, Lower bounds for constant weight codes, IEEE Trans. Info. Theory, Vol. IT.26 (1980)37-43; M.R. Best, Binary codes with a minimum distance of four, loc cit, Vol. IT- 26(1980), 738-743 and other papers in this journal On page 682, line 6, the value of X corresponding to F= 30 should be changed from 039 to 093 【实例截图】
【核心代码】
This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings,
C North-Holland Publishing Company 1977 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner L CC Number: 76-41296 ISBN:0444850090 and0444850104 Published by North-Holland Publishing Company Amsterdan· New york· Oxford Sole distributors for the U.S.A. and Canada Elsevier/North-Holland Inc 52 Vanderbilt Avenue New York. NY 10017 Library of Congress Cataloging in Publication Data Mac Williams, Florence Jessie. 1917 The theory of error-correcting codes 1. Error-correcting codes (Information theory) Sloane, Neil James Alexander, 1939- joint author IL. Title QA268.M3 5194 76-41296 ISBN0444850090 and0444850104 Second printing 1978 Third printing 1981 PRINTED IN THE NETHERLANDS P reface Coding theory began in the late 1940s with the work of Golay, Hamming and Shannon. Although it has its origins in an engineering problem, the subject has developed by using more and more sophisticated mathematical techniques. It is our goal to present the theory of error-correcting codes in a simple, easily understandable manner, and yet also to cover all the important aspects of the subject. Thus the reader will find both the simpler families of codes-for example, Hamming, BCH, cyclic and Reed-Muller codes-dis- cussed in some detail, together with encoding and decoding methods, as well as more advanced topics such as quadratic residue, Golay, Goppa, alternant, Kerdock, Preparata and self-dual codes and association schemes Our treatment of bounds on the size of a code is similarly thorough We discuss both the simpler results-the sphere-packing, Plotkin, Elias and Garshamov bounds-as well as the very powerful linear programming method and the McEliece -Rodemich-Rumsey-Welch bound, the best asymptotic result known. An appendix gives tables of bounds and of the best codes presently known of length up to 512 Having two authors has helped to keep things simple: by the time we both understand a chapter, it is usually transparent. Therefore this book can be used both by the beginner and by the expert, as an introductory textbook and as a reference book, and both by the engineer and the mathematician. of course this has not resulted in a thin book and so we suggest the following meilun An elementary first course on coding theory for mathematicians: Ch 1, Ch 2(56 up to Theorem 22), Ch 3, Ch 4(381-5), Ch 5(to Problem 5), Ch 7(not §87,8),Ch.8(§§1-3),Ch.9(§81,4),Ch.12(§8),Ch.13(§§1-3),Ch.14 (§§1-3) A second course for mathematicians:Ch.2(§§1-6,8),Ch.4(§6,7and art of 8), Ch 5(to Problem 6, and $83, 4, 5, 7), Ch 6(S$1-3, 10, omitting the Preface proof of Theorem33),Ch.8(§5,6),Ch.9(§§2,3,5),Ch.10(§§1-5,11),Ch.11, Ch. 13(584, 5,9),Ch. 16(891-6), Ch 17(87, up to Theorem 35), Ch. 19(SS1 An elementary first course on coding theory for engineers: Ch. 1, Ch.3, Ch.4(§§1-5),Ch.5( to problem5),Ch.7(not§7),Ch.9(§51,4,6),Ch.10 (§§1,2,5,6,7,10),Ch.13(§§1-3,6,7),Ch.14(§§1,2,4) A second course for engineers: Ch. 2(3$1-6), Ch. 8(5$1-3, 5, 6), Ch. 9 (S§2,3,5),Ch.10(§11),Ch.12(s1-3,8,9),Ch.l6(§§1,2,4,6,9),Ch.17(7, up to Theorem 35). There is then a lot of rich food left for an advanced course the rest of Chapters 2, 6, 1 l and 14, followed by Chapters 15, 18, 19, 20 and 21-a feast The following are the principal codes discussed Alternant Ch, 12 BCH,Ch.3,§S1,3;Ch.7,§6;Ch.8,§5;Ch.9;Ch.21,§8 Chien-Choy generalized BCH, Ch. 12, $7 Concatenated,Ch.10,§1l;Ch.18,§§5,8; Conference matrix, Ch 2, 84 Cyclic, Ch. 7, Ch. 8 Delsarte-Goethals, Ch. 15, $5 Difference-set cyclic, Ch. 13, $8 Double circulant and quasi-cyclic, Ch 16, $86-8 Euclidean and projective geometry, Ch. 13, $8 Goethals generalized Preparata, Ch. 15, 87 Golay(binary), Ch. 2, 86; Ch. 16, 82; Ch. 20 Golay (ternary), Ch. 16, 82: Ch. 20 Goppa,Ch.12,§83-5 Hadamard, Ch. 2. $3 Hamming,Ch.1,§7,Ch.7,§3 and Problem8; Irreducible or minimal cyclic,Ch.8,§§3,4; Justesen,Ch.10,§ll; Kerdock,Ch.2,§8;Ch.15,§5; Maximal distance separable, Ch. I1 Nordstrom- Robinson,Ch.2,§8;Ch.15,§§5,6 Pless symmetry, Ch. 16,88 Preparata,Ch2,§8;Ch.15,$6;Ch.18,§7.3; Product,Ch.18,§§2-6; Quadratic residue, Ch. 16 Redundant residue, Ch. 10,89 Reed-Muller, Ch. 1, $9 Chs. 13-15 Reed-Solomon, Ch. 10 Preface v Self-dual, Ch. 19 Single-error-correcting nonlinear, Ch. 2, 87; Ch. 18, 87.3 Srivastava, ch. 12, 86 Encoding methods are given for Linear codes, ch.1,82: Cyclic codes, Ch.7, 88 Reed-Solomon codes, Ch. 10, $7 Reed- Muller codes,Ch.13,§§6,7;Ch.14,§4. Decoding methods are given for Linear codes, Ch. 1, s83,4 Hamming codes, Ch. 1, 87 BCH codes, Ch3, 83; Ch9, $6: Ch. 12, 59; Reed-Solomon codes, Ch.10.$10 Alternant (including BCH, Goppa, Srivastava and Chien-Choy generalized BCH codes)Ch.12,§9 Quadratic residue codes, ch. 16, 89; Cyclic codes, Ch. 16, 89 while other decoding methods are mentioned in the notes to ch. 16 When reading the book, keep in mind this piece of advice, which should be given in every preface: if you get stuck on a section, skip it, but keep reading! Dont hesitate to skip the proof of a theorem: we often do. Starred sections are difficult or dull, and can be omitted on the first (or even second) reading. The book ends with an extensive bibliography. Because coding theory overlaps with so many other subjects (computers, digital systems, group theory, number theory, the design of experiments, etc. )relevant papers may be found almost anywhere in the scientific literature. Unfortunately this means that the usual indexing and reviewing journals are not always helpful We have therefore felt an obligation to give a fairly comprehensive bi biography. The notes at the ends of the chapters give sources for the theorems, problems and tables, as well as small bibliographies for some of the topics covered (or not covered) in the chapter Only block codes for correcting random errors are discussed we say little about codes for correcting other kinds of errors(bursts or transpositions )or about variable length codes convolutional codes or source codes (see the Preface otes to Ch. 1). Furthermore we have often considered only binary codes which makes the theory a lot simpler. most writers take the opposite point of view: they think in binary but publish their results over arbitrary fields There are a few topics which were included in the original plan for the book but have been reluctantly omitted for reasons of space: Gray codes and snake -in-the- box codes -see adelson et al. [5, 6] Buchner [210], Cavior [2531, Chien et al. [2901, Cohn [2991, Danzer and Klee 328], Davies [335], Douglas [382, 383], Even [4131, Flores [432], Gardner [4681, Gilbert [4811, Guy [571], Harper [605], Klee [764-767l, Mecklenberg et al. [951], Mills [9561, Preparata and Nievergelt [1083], Singleton [1215], Tang and Liu [1307], Vasilev [1367], Wyner [1440] and Yuen [1448, 1449] (ii Comma-free codes-see Ball and Cummings [60, 61, Baumert and Cantor [85], Crick et al. [3161, Eastman [399, golomb [523, pp. 118-122], Golomb et al. [5281, Hall [587, pp. 11-12, Jiggs [692], Miyakawa and moriya [967, Niho [992] and Redinbo and Walcott [1102]. See also the remarks on codes for synchronizing in the Notes to Ch. 1 (il) Codes with unequal error protection -see Gore and Kilgus [549] Kilgus and Gore [761] and Mandelbaum [9011 (iv) Coding for channels with feed back -see Berlekamp [1241, Horstein 664] and Schalkwijk et al. [1153-1155] v) Codes for the Gaussian channel-see Biglieri et al. [148-151], Blake [155, 156, 1581, Blake and Mullin [162], Chadwick et al. [256, 257, Gallager [4641, Ingemarsson [6831, Landau [7911, Ottoson [1017], Shannon [11911, Slepian [1221-1223] and Zetterberg [1456 (vi) The complexity of decoding-see Bajoga and Walbesser [59], Chaitin [257a-258a1, Gelfand et al. [471], Groth [564], Justesen [7061, Kolmogorov [774a1, Marguinaud [9161, Martin-Lof [917a], Pinsker [1046a], Sarwate [11451 and Savage [1149-1152a vi) The connections between coding theory and the packing of equal spheres in n-dimensional euclidean space-see Leech [803-805],[807, Leech and Sloane [808-810 and Sloane [1226 The following books and monographs on coding theory are our predeces sors: Berlekamp [113, 116], Blake and Mullin [162], Cameron and van lint [234, Golomb [522], Lin [8341, Van Lint [848], Massey [922a], Peterson [1036a, Peterson and Weldon [10401, Solomon [1251] and sloane [1227a]; while the following collections contain some of the papers in the bibliography Berlekamp [126], Blake [157, the special issues [377a, 678, 679], Hartnett [620], Mann [909]and Slepian [1224. see also the bibliography [1022 We owe a considerable debt to several friends who read the first draft very carefully, made numerous corrections and improvements, and frequently saved us from dreadful blunders. In particular we should like to thank IF Blake, P. Delsarte, J -M. Goethals, R.L. Graham, J.H. van Lint, G. Longo C L. Mallows, J. McKay, V. Pless, H.O. Pollak, L D. Rudolph, D.W. Sar wate, many other colleagues at Bell Labs, and especially A m. odlyzko for Preface their help. Not all of their suggestions have been followed, however, and the authors are fully responsible for the remaining errors. ( This conventional remark is to be taken seriously. ) We should also like to thank all the typists at Bell Labs who have helped with the book at various times, our secretary Peggy van Ness who has helped in countless ways, and above all Marion Messersmith who has typed and retyped most of the chapters Sam lomonaco has very kindly helped us check the galley proofs Preface to the third printing We should like to thank many friends who have pointed out errors and misprints. The corrections have either been made in the text or are listed below A Russian edition was published in 1979 by Svyaz(Moscow), and we are extremely grateful to L. A. Bassalygo, I. T. Grushko and v. a. Zinov'ev for producing a very careful translation. They supplied us with an extensive list of corrections. They also point out (in footnotes to the russian edition a number of places we did not cite the earliest source for a theorem we have corrected the most glaring omissions, but future historians of coding theory should also consult the russian edition Problem 17, page 75. Shmuel Schreiber has pointed out that not all ways of choosing the matrices A, B, C, d work. one choice which does work is 0001 0010 0100 1000 A=/1000 0100 0010 0010 000 1000 D=/007 0100 0100 1000 0001」 010 Page 36, notes to 82. Add after Wu [1435, 1436: K. sh. Zigangirov, Some sequential decoding procedures, Problems of Information Transmission, 2(4) (1966)1-10; and K Sh Zigangirov, Sequential Decoding Procedures(Svyaz, Moscow, 1974) Page 72, Research problem 2. 4. It is now known that A(10, 4)=40,72s A(11, 4)<79, and 144= A(12, 4)s158. See M.R. Best, Binary codes with a minimum distance of four, IEEE Trans. Info. Theory, Vol. IT- 26(6)(November 1980),738-743 Page 123, Research problem 4.1. Self-contained proofs have been given by O Moreno, On primitive elements of Trace 1 in GF(2m), Discrete Math., to appear and L. R. vermani, Primitive elements with nonzero trace, preprint Preface to the third printing Chapter 6, pp. 156 and 180. Theorem 33 was proved independently by Zinov'ev and Leont'ev [1472] Page 166, Research problem 6. I has been settled in the affirmative by I. I Dumer, A remark about codes and tactical configurations, Math. notes, to appear;and by C Roos, Some results on t-constant codes, IEEE Trans. Info Theory, to appear Page 175. Research problem 6. 3 has also been solved by dumer and roos [op cit Page 178-179 Research problems 6.4 and 6.5 have been solved by Dumer [op cit]. The answer to 6.5 is No Page 267, Fig9.1.R.E. Kibler(private communication) has pointed out that the asterisks may be removed from the entries [127, 29, 43], [255, 45, 87]and [255, 37,91] since there are minimum weight codewords which are low-degree multiples of the generator polynomial Page 280, Research problem 9.4. T Helleseth(IEEE Trans. Info. Theory, Vol IT.25( 1979)361-362)has shown that no other binary primitive bCh codes are quasi-perfect Page 299, Research problem 10.1. R.E. Kibler(private communication)has found a large number of such codes Page 323. The proof of Theorem 9 is valid only for g= 2m. For odd g the code need not be cyclic. See G. Falkner, w. Heise, B. Kowol and E Zehender, on the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Universita di modena, to appear. Page 394, line 9 from the bottom. As pointed out by Massey in[918, P. 100], the number of majority gates required in an L-step decoder need never exceed the dimension of the code Page 479. Research problem 15.2 is also solved in I. I. Dumer, Some new uniformly packed codes, in: Proceedings MFTI, Radiotechnology and elec- tronics Series(MFTI, Moscow, 1976)pp. 72-78 Page 546. The answer to Research problem 17.6 is No, and in fact R. E Kibler Some new constant weight codes, IEEE Trans. Info. Theory, Vol. IT-26(May 1980)364-365, hows that27≤A(24,l0,8)≤68 Appendix a, Figures I and 3. For later versions of the tables of A(n, d)and A(n, d, w)see R. L. Graham and n.j. A Sloane, Lower bounds for constant weight codes, IEEE Trans. Info. Theory, Vol. IT.26 (1980)37-43; M.R. Best, Binary codes with a minimum distance of four, loc cit, Vol. IT- 26(1980), 738-743 and other papers in this journal On page 682, line 6, the value of X corresponding to F= 30 should be changed from 039 to 093 【实例截图】
【核心代码】
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