Table Of ContentSignals and Communication Technology
Giovanni Cancellieri
Polynomial
Theory of Error
Correcting
Codes
Signals and Communication Technology
More information about this series at http://www.springer.com/series/4748
Giovanni Cancellieri
Polynomial Theory
of Error Correcting Codes
123
GiovanniCancellieri
InformationEngineering
Polytechnic University ofMarche
Ancona
Italy
ISSN 1860-4862 ISSN 1860-4870 (electronic)
ISBN 978-3-319-01726-6 ISBN 978-3-319-01727-3 (eBook)
DOI 10.1007/978-3-319-01727-3
LibraryofCongressControlNumber:2014947130
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To Daniele, my son and pride
Preface
Error correcting codes represent a widely applied technique for assuring reliable
electronic communications and data recording. Although often some codes are
introduced without the need for a polynomial approach, the attempt to provide a
descriptionwherepolynomialtheoryishoweverpresentcould beofsomeinterest.
The intent of this book is to develop such theory, in a unitary way, trying to find
severalconceptualbridgesbetweendifferentclassesofcodes(block,convolutional,
concatenated,…). This goal requires the introduction of some rather unusual
mathematical tools (like interleaved polynomial multiplication or interleaved
polynomial division), able to support it. Also some structural transformations, like
modified code lengthening or modified H-extension, are needed in order to con-
struct a coherent model, able to support noticeable new interpretations.
Theintroductionofquasi-cycliccodesasatruegeneralizationoftheconceptof
cyclic codes is only an example of the fruitful use of such mathematical tools and
geometrical transformations. They are also important for giving an intuitive justi-
ficationfortheencodercircuitstobeadopted.Statediagrams,constructedonthese
encoder circuits, contribute to give a better understanding of the properties char-
acterizing the codes under study, besides the measure of the decoding computa-
tional complexity.
The distinction between well-designed and not well-designed convolutional
codes represents another innovative concept. The latter family of codes is equiv-
alent to catastrophic convolutional codes, but since they are systematic, the cata-
strophic behavior is no longer a problem. On the other hand, a more than linear
increaseinthenumberoflow-weightcodeframeswiththenumberofframeperiods
remainsadrawbackfornotwell-designedconvolutionalcodes,togetherwithsome
difficulties in their parity check matrix determination and tail-biting arrangement
organization.Directproductcodesbetweenapairofblockcodesaredemonstrated
to be a subset of not well-designed convolutional codes. Some further conceptual
(and rather surprising) bridges between block codes and convolutional codes are
constructed.
The treatment is organized maintaining distinct the approaches based on the
generator matrix and on the parity check matrix. The reader is gradually guided to
vii
viii Preface
theinterpretationofsuchviewpoints.Someconcepts,derivingfromdualproperties,
willappearinalltheirstrongefficacyonlyatthismoment.Modifiedlengtheningof
cyclic codes and modified H-extension of cyclic codes are dual opportunities
leadingtoanewcomprehensionoftheintrinsicnatureofconvolutionalcodes.The
former is obtained by acting on the generator matrix, and the latter on the parity
checkmatrix.Thesamecanbemadeforquasi-cycliccodes.Propertransformation
of control symbols into information symbols and vice versa supports the above
modifications.
Modern coding (mainly regarding turbo codes and LDPC codes) is a topic,
which is faced after a wide application of the above innovative concepts, so
allowing a comprehensive understanding of the structures characterizing such
codes. For instance, modified H-extension of quasi-cyclic codes offers the possi-
bilityofsettingupasortofdoublyconvolutionalLDPCcode,inturninterpretedin
relationtoproperturboproductschemes.TheintroductionofBPGcodes(Binomial
Product Generator codes) allows to treat array LDPC codes and some forms of
concatenated LDPC codes, together with their convolutional versions, not only by
means of the parity check matrix.
The continuous search for ultimate extremely good performance, present in the
recent literature, has the consequence of reducing attention to the above theoretic
aspects. The dominant use of computer simulations and optimization procedures
often entails a nonexhaustive investigation of the true geometrical nature of the
code under study. Many code families, apparently very different, on more careful
analysis, would appear instead strictly related. All these considerations may pro-
duce interesting future theoretical developments. To this purpose, the present
comprehensive approach would give a possible contribution.
About 250 Definitions, 300 propositions (Theorems, Corollaries, Lemmas),
nearly 500 examples, highly interconnected, can give an idea of the amount of
contentstreated.InfourAppendicessomeusefulconcepts,notstrictlyrelatedtothe
topics under development, are collected. They are devoted to nonskilled readers,
who need auxiliary assistance for framing such theory in a proper context.
Acknowledgments
I wish to thank Marco Baldi for fruitful discussions and continuous help in
obtainingmostpartofthenumerical computations,FrancoChiaraluceforhishints
and theoretical support, Andrea Carassai for some intuitive views, my students at
the Polytechnic University of Marche for their contribution in the organization of
some material, and finally Laura Fonti for typewriting help and figure assembling.
AparticularacknowledgmentisdevotedtoMarcoChianiforhavingreviewedpart
of the content, when it was under development. I am also indebted to Rossano
Marchesani, Roberto Garello, Torleiv Klove, for our collaboration during the long
research activity at the basis of the present work.
Contents
Part I Generator Matrix
1 Generator Matrix Approach to Linear Block Codes. . . . . . . . . . . 3
1.1 Additive n × n Linear Transformation
of a Binary Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Generator Matrix G of a Linear Block Code. . . . . . . . . . . . . 5
1.3 Polynomial Description of the Generator Matrix
in a Linear Block Code. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Properties of a Linear Block Code Derived
from the Structural Characteristics of g(x) . . . . . . . . . . . . . . 15
1.5 Systematic Encoder Circuit. . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Code Concatenation: Effects on G Matrix . . . . . . . . . . . . . . 21
1.7 Code Puncturation: Effects on G Matrix. . . . . . . . . . . . . . . . 28
1.8 Cyclic Block Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.9 Enumeration of all the Possible Cyclic Codes of Length N. . . 37
1.10 Shortened Cyclic (SC) Codes. . . . . . . . . . . . . . . . . . . . . . . 44
1.11 Lengthened Cyclic (LC) Codes. . . . . . . . . . . . . . . . . . . . . . 47
1.12 Subcode of an s.s. Time-Invariant Polynomial Code . . . . . . . 56
1.13 Modified Lengthened Cyclic (MLC) Codes . . . . . . . . . . . . . 58
1.14 State Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.15 Direct Product Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
1.16 Generator Matrix of a Direct Product Code . . . . . . . . . . . . . 73
1.17 Direct Product Codes as MLC Codes . . . . . . . . . . . . . . . . . 75
1.18 Interpretation of Particular Direct Product Codes
by Means of GPC Codes. . . . . . . . . . . . . . . . . . . . . . . . . . 76
1.19 Cyclic and Pseudo-Cyclic Codes in a Non-binary
Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
1.20 Q-ary State Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.21 Main Families of Non-binary Block Codes. . . . . . . . . . . . . . 85
ix
x Contents
1.22 Reed-Solomon Codes and Other MDS Non-binary Codes . . . 89
1.23 Trellis for an s.s. Time-Invariant Block Code Obtained
from Its Generator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 95
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2 Wide-Sense Time-Invariant Block Codes in Their
Generator Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.1 Periodically Time-Varying Generator Matrix . . . . . . . . . . . . 101
2.2 Quasi-Cyclic Codes (QC) as a Widening in the Concept
of Cyclic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.3 Quasi-Cyclic Codes with Distributed Control Symbols
Described with Their G Matrix. . . . . . . . . . . . . . . . . . . . . . 112
2.4 Representation of Known Block Codes as QC Codes
with Distributed Control Symbols. . . . . . . . . . . . . . . . . . . . 117
2.5 Relation Between Some Binary QC-Codes and Cyclic
or Pseudo-Cyclic Codes in a Q-Ary Alphabet. . . . . . . . . . . . 119
2.6 Encoder Circuits Based on the G Matrix for a QC Code . . . . 121
2.7 Shortened Quasi-Cyclic (SQC) Codes . . . . . . . . . . . . . . . . . 125
2.8 Lengthened Quasi-Cyclic (LQC) Codes. . . . . . . . . . . . . . . . 130
2.9 Subcode of a w.s. Time-Invariant Polynomial Code. . . . . . . . 136
2.10 Modified Lengthened Quasi-Cyclic (MLQC) Codes. . . . . . . . 138
2.11 Trellis for a w.s. Time-Invariant Block Code Obtained
from Its Generator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 143
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3 Generator Matrix Approach to s.s. Time-Invariant Convolutional
Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.1 Traditional View of Non-systematic s.s. Time-Invariant
Convolutional Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.2 State Diagram and Minimum Distance. . . . . . . . . . . . . . . . . 154
3.3 Systematic Convolutional Codes. . . . . . . . . . . . . . . . . . . . . 162
3.4 Low-Rate Convolutional Codes . . . . . . . . . . . . . . . . . . . . . 166
3.5 High-Rate Punctured Convolutional Codes. . . . . . . . . . . . . . 172
3.6 Recursive Systematic Convolutional (RSC) Codes. . . . . . . . . 176
3.7 Equivalence Between MLC Codes and s.s.
Time-Invariant Convolutional Codes . . . . . . . . . . . . . . . . . . 180
3.8 Strict-Sense Time-Invariant High-Rate Convolutional (MLC)
Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.9 A First Bridge Between Cyclic Block Codes and s.s.
Time-Invariant Convolutional Codes . . . . . . . . . . . . . . . . . . 190
3.10 Tail-Biting s.s. Time-Invariant Convolutional Codes . . . . . . . 195
3.11 Trellis of an s.s. Time-Invariant Convolutional
Code Obtained from Its Generator Matrix . . . . . . . . . . . . . . 199
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
