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The art of error correcting codes
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The Art
of
Error
Correcting
Coding
The Art of Error Correcting Coding
Robert H. Morelos-Zaragoza
Copyright © 2002 John Wiley & Sons Ltd
ISBNs: 0-471-49581-6 (Hardback); 0-470-84782-4 (Electronic)
The Art of
Correcting
Coding
Robert
H.
Morelos-Zaragoza
SONY Computer Science Laboratories,
Inc.
JAPMV
JOHN
WILEY
&
SONS,
LTD
Copyright
0
2002 by John Wiley
&
Sons, Ltd
Baffins Lane, Chichester,
West
Sussex, PO19 IUD, England
National 01243 779777
International
(+44) 1243 779777
e-mail (for orders and customer service enquiries): cs-books@viley.co.uk
Visit our Home Page on
http://www.wileyeurope.com
or http://www.wiley.com
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, scanning or otherwise, except
under the
term of the Copyright Designs and Patents Ad 1988 or under the terms of a licence issued by the
Copyright Licensing Agency,
90
Tonenham Court Road, London, W1P 9HE, UK, without the permission in writing of the Publisher, with the
exception of any material supplied specifically for the purpose of being entered and executed on a computer sys
-
tem, for exdusive use by the purchaser of the publication.
Neither the
author(s) nor John Wiley
&
Sons, Ltd accept any responsibility or liability for loss or damage occa
-
sioned to any person or property through using the material, instructions, methods or ideas contained herein, or
acting or refraining from acting as a result of such use. The
author(s) and Publisher expressly disclaim all implied
warranties, including merchantability of fitness for any particular purpose. There will be no duty on the
author(s)
or Publisher to correct any errors or defects in the software.
Designations used by companies to distinguish their products are often claimed as trademarks. In all instances
where John Wiley
&
Sons, Ltd is aware of a daim, the product names appear in initial capital or capital letters.
Readers, however, should contact the appropriate companies for more complete information regarding trade
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marks and registration.
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Why
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John Wiley
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Sons, Inc., 605 Third Avenue,
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British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0471 49581 6
Produced from
LaTeX files supplied
by
the author.
Printed and bound in Great Britain by
Antony Rowe Ltd, Chippenham, Wiltshire.
This book is printed on acid
-
free paper responsibly manufactured from sustainable forestry, in which at least two
trees are planted for each one used for paper production.
Contents
Preface
..........................................
ix
.........................................
Foreword
xi
...
....................................
The
ECC
web site
xi11
.....................................
1
Introduction
....................
1.1
Error correcting coding: Basic concepts
1.1.1 Block codes and convolutional codes
.................
1.1.2
Hamming distance, Hamming spheres and error correcting capability
...............................
1.2 Linear block codes
..................
1.2.1 Generator and parity
-
check matrices
1.2.2 The weight is the distance
.......................
.................
1.3
Encoding and decoding of linear block codes
........................
1.3.1 EncodingwithGandH
........................
1.3.2 Standard array decoding
1.3.3 Hamming spheres, decoding regions and the standard array
......
...................
1.4
Weight distribution and error performance
1.4.1 Weight distribution and undetected error probability over a BSC
...
1.4.2 Performance bounds over BSC
.
AWGN and fading channels
.....
..........
1.5
General structureof a
hard-decisiondecoder of
linearcodes
.....................
2
Hamming. Golay and Reed
-
Muller codes
23
.................................
2.1 Hamming codes 23
..................
2.1.1 Encoding and decoding procedures 24
.............................
2.2 The binary Golay code
25
................................
2.2.1 Encoding 25
................................
2.2.2 Decoding 26
......
2.2.3 Arithmetic decoding of the extended (24.
12.
8) Golay code
26
...........................
2.3 Binary Reed
-
Muller codes 27
..................
2.3.1 Boolean polynomials and
RM
codes 27
............
2.3.2 Finite geometries and majority
-
logic decoding 28
........................
3
Binary cyclic codes and
BCH
codes
33
...............................
3.1 Binary cyclic codes 33
................
3.1.1 Generator and parity
-
check polynomials 33
CONTENTS
.......................
3.1.2 The generator polynomial 34
.............
3.1.3 Encoding and decoding of binary cyclic codes 35
......................
3.1.4 The parity
-
check polynomial 36
................
3.1.5 Shortened cyclic codes and CRC codes 37
........................
3.2 General decoding of cyclic codes 39
...........................
3.2.1
GF(2m)
arithmetic 41
...............................
3.3 Binary BCH codes
44
................................
3.4 Polynomial codes 45
.........................
3.5 Decoding of binary BCH codes 46
..............
3.5.1 General decoding algorithm for BCH codes 48
...............
3.5.2 The Berlekamp
-
Massey algorithm (BMA) 49
..............................
3.5.3
PGZ
decoder 52
.......................
3.5.4 Euclidean Algorithm (EA) 53
...................
3.5.5 Chien search and error correction 55
.....................
3.5.6 Errors
-
and
-
erasures decoding 55
..................
3.6 Weight distribution and performance bounds 56
.....................
3.6.1 Error performance evaluation 57
...................
4
Non
-
binary
BCH
codes: Reed
-
Solomon codes 63
.........................
4.1
RS
codes as polynomial codes 63
.........................
4.2 From binary
BCH
to RS codes 63
...............................
4.3 Decoding RS codes 64
....................
4.3.1 Remarks on decoding algorithms 69
.....................
4.3.2 Errors
-
and
-
erasures decoding 69
...............................
4.4 Weight distribution 73
5
Binaryconvolutionalcodes
.............................
75
.................................
5.1
Basic structure 75
5.1.1 Recursive systematic convolutional codes
...............
80
..............................
5.1.2 Free distance 81
.........................
5.2 Connections with block codes 81
..........................
5.2.1 Zero
-
tail construction 81
.....................
5.2.2 Direct
-
truncation construction 82
.........................
5.2.3 Tail
-
biting construction 82
5.2.4 Weight distributions
..........................
83
5.3 Weight enumeration and performance bounds
.................
84
5.4 Decoding: Viterbi algorithm with Hamming metrics
..............
86
5.4.1 Maximum likelihood decoding and metrics
..............
87
5.4.2 The Viterbi algorithm
..........................
88
.........................
5.4.3 Implementation issues 90
.........................
5.5 Punctured convolutional codes 96
5.5.1
Implementation issues related to punctured convolutional codes ... 99
..............................
5.5.2 RCPC codes 100
..........................
6
Modifying and combining codes 103
................................
6.1 Modifying codes 103
...............................
6.1.1
Shortening 103
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