Quaternary codes

248 Pages·3.227 MB·English

by  Zhe-Xian Wan.

QUATERNARY
CODES
SERIES ON APPLIED MATHEMATICS
Editor-in-Chief: Frank Hwang
Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum
Vol. 1 International Conference on Scientific Computation
eds. T. Chan and Z.-C. Shi
Vol. 2 Network Optimization Problems - Algorithms, Applications and
Complexity
eds. D.-Z. Du and P. M. Pandalos
Vol. 3 Combinatorial Group Testing and Its Applications
by D.-Z. Du and F. K. Hwang
Vol. 4 Computation of Differential Equations and Dynamical Systems
eds. K. Feng and Z.-C. Shi
Vol. 5 Numerical Mathematics
eds. Z.-C. Shi and T. Ushijima
Vol. 6 Machine Proofs in Geometry
by S.-C. Chou, X.-S. Gao and J.-Z. Zhang
Vol. 7 The Splitting Extrapolation Method
by C. B. Liem, T. LQ and T. M. Shih
Vol. 8 Quaternary Codes
by z.-x. Wan
Series on
Applied Mathematics
VolumeS
UATERNARY
CODES
Zhe-Xian Wan
Chinese Academy of Sciences, China
and
Lund University, Sweden
b
l
World Scientific
II
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Library of Congress Cataloging-in-Publication Data
Wan, Zhe-Xian
Quaternary codes / by Zhe-Xian Wan.
p. cm. --(Series on applied mathematics; v. 8)
Includes bibliographical references and index.
ISBN 9810232748 (alk. paper)
I. Coding theory. I. Title. II. Series.
QA268.W35 1997
003'.54--dc21 97-29178
CIP
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PREFACE
A binary error-correcting code of length n is just a subset of the vector
space F2' and linear codes are subspaces of F2'. The vectors in a code are
called codewords and the Hamming distance between two codewords is the
number of positions in which they differ. The rate of a code of length n is
defined to be the logarithm to the base 2 of the number of codewords in the
code divided by n. One of the fundamental problems in coding theory is to
construct and study codes of length n with large rate subject to the condition
that the minimum of the distances between any two different codewords is
some given integer d, the minimal distance of the code.
Historically, linear codes have been the most important codes since they
are easier to construct, encode, and decode. Around 1970 several binary non
linear codes having at least twice as many codewords as any linear code with
the same length and minimal distance have been constructed. Among them
are the Nordstrom-Robinson code, the Preparata codes, the Kerdock codes,
the Goethals codes, the Delsarte-Goethals codes, etc. However, these binary
nonlinear codes are not so easy to describe, to encode and decode as the linear
codes. It is also discovered that the weight enumerator of the Preparata code
is the MacWilliams transform of that of the Kerdock code of the same length,
though they are not dual to each other, which seems to be a mystery in coding
theory.
A surprising breakthrough in coding theory is that the Kerdock codes can
be viewed as cyclic codes over Z4 (Nechaev (1989) and Hammons et ai. (1994))
and the binary image of the Z4-dual of the Kerdock code over Z4 can be
regarded as a variant of the Preparata code (Hammons et al. (1994)). This
leads to a new direction in coding theory, the study of cyclic codes over Z4.
This book aims to be an introduction to this new direction. The first draft
was prepared for several lectures at the Department of Mathematics, Shaanxi
Normal University, Xi'an, China in May 1996 and the second draft for a series
VII
viii Preface
of lectures at the Department of Information Technology, Lund University,
Lund, Sweden. Then these drafts were revised completely to the present form.
The Hensel lemma and Galois rings which are important tools for the study
of Z4-codes are included. The Gray map being a connection between Z4-codes
and their binary images is introduced. The quaternary Kerdock codes and
Preparata codes and their binary images are studied in detail. The construction
of lattices from Z4-codes and the weight enumerators of self-dual Z4-codes are
mentioned. To read the book only a rudiment of binary codes is necessary.
The author is indebted to Rolf Johannesson who supported the author's
work in many aspects and created an active and productive atmosphere in the
Information Theory Group in Lund where the present book was written. The
author is also indebted to Anupama Pawar K. and Babitha Yadav for their
beautiful typesetting and to E. H. Chionh for her helpful and careful editorial
work. Without their support and help the book could not have appeared so
soon.
Zhe-Xian Wan
CONTENTS
Preface vii
1. Quaternary Linear Codes and Their Generator Matrices 1
1.1. Definition 1
1.2. Generator Matrices 4
1.3. Examples 7
2. Weight Enumerators 9
2.1. Weight Enumerators of Quaternary Codes 9
2.2. Krawtchouk Polynomials 18
2.3. Distance Enumerators of Binary Codes 26
3. The Gray Map 35
3.1. The Gray Map 35
3.2. Binary Images of Z4-Codes 38
3.3. Linearity Conditions 44
3.4. Binary Codes Associated with a Z4-Linear Code 48
4. Z4-Linearity and Z4-Nonlinearity of Some Binary
Linear Codes 53
4.1. A Review of Reed-Muller Codes 53
4.2. The Z4-Linearity of Some RM(r, m) 55
4.3. The Z4-Nonlinearity of Extended Binary Hamming
Codes H2~ when m :::: 5 57
5. Hensel's Lemma and Hensel Lift 63
5.1. Hensel's Lemma 63
5.2. Basic Irreducible Polynomials 66
5.3. Some Concepts from Commutative Ring Theory 68
5.4. Factorization of Monic Polynomials in Z4[X] 70
5.5. Hensel Lift 73
6. Galois Rings 77
6.1. The Galois Ring GR(4ffi 77
)
IX
x Contents
6.2. The 2-Adic Representation 81
6.3. Automorphisms of GR(4m) 85
6.4. Basic Primitive Polynomials Which Are Hensel Lifts 88
6.5. Dependencies among ~j 90
7. Cyclic Codes 93
7.1. A Review of Binary Cyclic Codes 93
7.2. Quaternary Cyclic Codes 96
7.3. Sun Zi Theorem 98
7.4. Ideals in Z4[Xl/(Xn - 1) 104
8. Kerdock Codes 113
8.1. The Quaternary Kerdock Codes 113
8.2. Trace Descriptions of K(m) 116
8.3. The Kerdock Codes 121
8.4. Weight Distributions of the Kerdock Codes 126
8.5. Soft-Decision Decoding of Quaternary Kerdock Codes 130
9. Preparata Codes 133
9.1. The Quaternary Preparata Codes 133
9.2. The "Preparata" Codes 139
9.3. Decoding P(m) in the Z4-Domain 142
9.4. The Preparata Codes 145
10. Generalizations of Quaternary Kerdock and
Preparata Codes 155
10.1. Quaternary Reed-Muller Codes 155
10.2. Quaternary Goethals Codes 163
10.3. Quaternary Delsarte-Goethals and Goethals-Delsarte Codes 170
10.4. Automorphism Groups 171
11. Quaternary Quadratic Residue Codes 177
11.1. A Review of Binary Quadratic Residue Codes 177
11.2. Quaternary Quadratic Residue Codes 184
12. Quaternary Codes and Lattices 195
12.1. Lattices 195
12.2. A Construction of Lattices from Quaternary Linear Codes 198
13. Some Invariant Theory 205
13.1. The Poincare Series 205
13.2. Molien's Theorem 207
13.3. Hilbert's Finite Generation Theorem 212