Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

527 Pages·2017·9.24 MB·English

by  Martin Tomlinson, Cen Jung Tjhai, Marcel A. Ambroze, Mohammed Ahmed, Mubarak Jibril

Signals and Communication Technology
Martin Tomlinson
Cen Jung Tjhai
Marcel A. Ambroze
Mohammed Ahmed
Mubarak Jibril
Error-Correction
Coding and
Decoding
Bounds, Codes, Decoders, Analysis and
Applications
Signals and Communication Technology
More information about this series at http://www.springer.com/series/4748
Martin Tomlinson • Cen Jung Tjhai
Marcel A. Ambroze • Mohammed Ahmed
Mubarak Jibril
Error-Correction Coding
and Decoding
Bounds, Codes, Decoders, Analysis
and Applications
Martin Tomlinson Mohammed Ahmed
School of Computing, Electronics School of Computing, Electronics
and Mathematics and Mathematics
Plymouth University Plymouth University
Plymouth, Devon Plymouth, Devon
UK UK
Cen Jung Tjhai Mubarak Jibril
PQ Solutions Limited Satellite Applications and Development
London Nigeria Communications Satellite Limited
UK Abuja
Nigeria
Marcel A. Ambroze
School of Computing, Electronics
and Mathematics
Plymouth University
Plymouth, Devon
UK
ISSN 1860-4862 ISSN 1860-4870 (electronic)
Signals and Communication Technology
ISBN 978-3-319-51102-3 ISBN 978-3-319-51103-0 (eBook)
DOI 10.1007/978-3-319-51103-0
Library of Congress Control Number: 2016963415
© The Editor(s) (if applicable) and The Author(s) 2017. This book is published open access.
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Preface
The research work described in this book is some of the works carried out by the
authors whilst working in the Coding Group at the University of Plymouth, U.K.
The Coding Group consists of enthusiastic research students, research and teaching
staff members providing a very stimulating environment to work. Also being driven
by academic research, a significant number of studies were driven by the com-
munications industry with their many varying applications and requirements of
error-correcting codes. This partly explains the variety of topics covered in this
book.
Plymouth, UK Martin Tomlinson
London, UK Cen Jung Tjhai
Plymouth, UK Marcel A. Ambroze
Plymouth, UK Mohammed Ahmed
Abuja, Nigeria Mubarak Jibril
vii
Acknowledgements
We would like to thank all of our past and present research students, our friends and
fellow researchers around the world who have helped our understanding of this
fascinating and sometimes tricky subject. Special thanks go to our research col-
laborators Des Taylor, Philippa Martin, Shu Lin, Marco Ferrari, Patrick Perry, Mark
Fossorier, Martin Bossert, Eirik Rosnes, Sergey Bezzateev, Markus Grassl,
Francisco Cercas and Carlos Salema. Thanks also go to Dan Costello, Bob
McEliece, Dick Blahut, David Forney, Ralf Johannason, Bahram Honary, Jim
Massey and Paddy Farrell for interesting and informed discussions. We would also
like to thank Licha Mued for spending long hours editing the manuscript.
ix
Contents
Part I Theoretical Performance of Error-Correcting Codes
1 Bounds on Error-Correction Coding Performance . . . . . . . . . . . . . 3
1.1 Gallager’s Coding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Linear Codes with a Binomial Weight Distribution . . . 7
1.1.2 Covering Radius of Codes . . . . . . . . . . . . . . . . . . . . . . 13
1.1.3 Usefulness of Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Bounds on the Construction of Error-Correcting Codes . . . . . . . 13
1.2.1 Upper Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.3 Lower Bounds from Code Tables . . . . . . . . . . . . . . . . . 21
1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Soft and Hard Decision Decoding Performance . . . . . . . . . . . . . . . . 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Hard Decision Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Complete and Bounded Distance Decoding . . . . . . . . . 26
2.2.2 The Performance of Codes on the Binary Symmetric
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Soft Decision Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Performance Assuming a Binomial Weight
Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Performance of Self-dual Codes . . . . . . . . . . . . . . . . . . 39
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Soft Decision and Quantised Soft Decision Decoding . . . . . . . . . . . . 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Soft Decision Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xi
xii Contents
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 A Hard Decision Dorsch Decoder and BCH Codes . . . . . . . . . . 53
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Part II Code Construction
4 Cyclotomic Cosets, the Mattson–Solomon Polynomial,
Idempotents and Cyclic Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Cyclotomic Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 The Mattson–Solomon Polynomial . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Binary Cyclic Codes Derived from Idempotents . . . . . . . . . . . . 73
4.4.1 Non-Primitive Cyclic Codes Derived from
Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Binary Cyclic Codes of Odd Lengths from 129 to 189 . . . . . . . 78
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Good Binary Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Algorithms to Compute the Minimum Hamming Distance
of Binary Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 The First Approach to Minimum Distance
Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 Brouwer’s Algorithm for Linear Codes . . . . . . . . . . . . 104
5.2.3 Zimmermann’s Algorithm for Linear Codes
and Some Improvements. . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.4 Chen’s Algorithm for Cyclic Codes . . . . . . . . . . . . . . . 107
5.2.5 Codeword Enumeration Algorithm . . . . . . . . . . . . . . . . 111
5.3 Binary Cyclic Codes of Lengths 129 n 189 . . . . . . . . . . . 114
5.4 Some New Binary Cyclic Codes Having Large Minimum
Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.5 Constructing New Codes from Existing Ones . . . . . . . . . . . . . . 118
5.5.1 New Binary Codes from Cyclic Codes
of Length 151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5.2 New Binary Codes from Cyclic Codes
of Length 199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.6 Concluding Observations on Producing New Binary Codes. . . . 124
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135