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1 Coding and Capacity1

1.1 Digital Data Communication and Storage1

1.2 Channel-Coding Overview3

1.3 Channel-Code Archetype：The（7，4）Hamming Code4

1.4 Design Criteria and Performance Measures7

1.5 Channel-Capacity Formulas for Common Channel Models10

1.5.1 Capacity for Binary-Input Memoryless Channels11

1.5.2 Coding Limits for M-ary-Input Memoryless Channels18

1.5.3 Coding Limits for Channels with Memory21

Problems24

References26

2 Finite Fields，Vector Spaces，Finite Geometries，and Graphs28

2.1 Sets and Binary Operations28

2.2 Groups30

2.2.1 Basic Concepts of Groups30

2.2.2 Finite Groups32

2.2.3 Subgroups and Cosets35

2.3 Fields38

2.3.1 Definitions and Basic Concepts38

2.3.2 Finite Fields41

2.4 Vector Spaces45

2.4.1 Basic Definitions and Properties45

2.4.2 Linear Independence and Dimension46

2.4.3 Finite Vector Spaces over Finite Fields48

2.4.4 Inner Products and Dual Spaces50

2.5 Polynomials over Finite Fields51

2.6 Construction and Properties of Galois Fields56

2.6.1 Construction of Galois Fields56

2.6.2 Some Fundamental Properties of Finite Fields64

2.6.3 Additive and Cyclic Subgroups69

2.7 Finite Geometries70

2.7.1 Euclidean Geometries70

2.7.2 Projective Geometries76

2.8 Graphs80

2.8.1 Basic Concepts80

2.8.2 Paths and Cycles84

2.8.3 Bipartite Graphs86

Problems88

References90

Appendix A92

3 Linear Block Codes94

3.1 Introduction to Linear Block Codes94

3.1.1 Generator and Parity-Check Matrices95

3.1.2 Error Detection with Linear Block Codes98

3.1.3 Weight Distribution and Minimum Hamming Distance of a Linear Block Code99

3.1.4 Decoding of Linear Block Codes102

3.2 Cyclic Codes106

3.3 BCH Codes111

3.3.1 Code Construction111

3.3.2 Decoding114

3.4 Nonbinary Linear Block Codes and Reed-Solomon Codes121

3.5 Product，Interleaved，and Concatenated Codes129

3.5.1 Product Codes129

3.5.2 Interleaved Codes130

3.5.3 Concatenated Codes131

3.6 Quasi-Cyclic Codes133

3.7 Repetition and Single-Parity-Check Codes142

Problems143

References145

4 Convolutional Codes147

4.1 The Convolutional Code Archetype147

4.2 Algebraic Description of Convolutional Codes149

4.3 Encoder Realizations and Classifications152

4.3.1 Choice of Encoder Class157

4.3.2 Catastrophic Encoders158

4.3.3 Minimal Encoders159

4.3.4 Design of Convolutional Codes163

4.4 Alternative Convolutional Code Representations163

4.4.1 Convolutional Codes as Semi-Infinite Linear Codes164

4.4.2 Graphical Representations for Convolutional Code Encoders170

4.5 Trellis-Based Decoders171

4.5.1 MLSD and the Viterbi Algorithm172

4.5.2 Differential Viterbi Decoding177

4.5.3 Bit-wise MAP Decoding and the BCJR Algorithm180

4.6 Performance Estimates for Trellis-Based Decoders187

4.6.1 ML Decoder Performance for Block Codes187

4.6.2 Weight Enumerators for Convolutional Codes189

4.6.3 ML Decoder Performance for Convolutional Codes193

Problems195

References200

5 Low-Density Parity-Check Codes201

5.1 Representations of LDPC Codes201

5.1.1 Matrix Representation201

5.1.2 Graphical Representation202

5.2 Classifications of LDPC Codes205

5.2.1 Generalized LDPC Codes207

5.3 Message Passing and the Turbo Principle208

5.4 The Sum-Product Algorithm213

5.4.1 Overview213

5.4.2 Repetition Code MAP Decoder and APP Processor216

5.4.3 Single-Parity-Check Code MAP Decoder and APP Processor217

5.4.4 The Gallager SPA Decoder218

5.4.5 The Box-Plus SPA Decoder222

5.4.6 Comments on the Performance of the SPA Decoder225

5.5 Reduced-Complexity SPA Approximations226

5.5.1 The Min-Sum Decoder226

5.5.2 The Attenuated and Offset Min-Sum Decoders229

5.5.3 The Min-Sum-with-Correction Decoder231

5.5.4 The Approximate Min* Decoder233

5.5.5 The Richardson/Novichkov Decoder234

5.5.6 The Reduced-Complexity Box-Plus Decoder236

5.6 Iterative Decoders for Generalized LDPC Codes241

5.7 Decoding Algorithms for the BEC and the BSC243

5.7.1 Iterative Erasure Filling for the BEC243

5.7.2 ML Decoder for the BEC244

5.7.3 Gallager’s Algorithm A and Algorithm B for the BSC246

5.7.4 The Bit-Flipping Algorithm for the BSC247

5.8 Concluding Remarks248

Problems248

References254

6 Computer-Based Design of LDPC Codes257

6.1 The Original LDPC Codes257

6.1.1 Gallager Codes257

6.1.2 MacKay Codes258

6.2 The PEG and ACE Code-Design Algorithms259

6.2.1 The PEG Algorithm259

6.2.2 The ACE Algorithm260

6.3 Protograph LDPC Codes261

6.3.1 Decoding Architectures for Protograph Codes264

6.4 Multi-Edge-Type LDPC Codes265

6.5 Single-Accurnulator-Based LDPC Codes266

6.5.1 Repeat -Accumulate Codes267

6.5.2 Irregular Repeat-Accumulate Codes267

6.5.3 Generalized Accumulator LDPC Codes277

6.6 Double-Accumulator-Based LDPC Codes277

6.6.1 Irregular Repeat -Accumulate-Accumulate Codes278

6.6.2 Accumulate-Repeat-Accumulate Codes279

6.7 Accurnulator-Based Codes in Standards285

6.8 Generalized LDPC Codes287

6.8.1 A Rate-1/2 G-LDPC Code290

Problems292

References295

7 Turbo Codes298

7.1 Parallel-Concatenated Convolutional Codes298

7.1.1 Critical Properties of RSC Codes299

7.1.2 Critical Properties of the Interleaver300

7.1.3 The Puncturer301

7.1.4 Performance Estimate on the BI-AWGNC301

7.2 The PCCC Iterative Decoder306

7.2.1 Overview of the Iterative Decoder308

7.2.2 Decoder Details309

7.2.3 Summary of the PCCC Iterative Decoder313

7.2.4 Lower-Complexity Approximations316

7.3 Serial-Concatenated Convolutional Codes320

7.3.1 Performance Estimate on the BI-AWGNC320

7.3.2 The SCCC Iterative Decoder323

7.3.3 Summary of the SCCC Iterative Decoder325

7.4 Turbo Product Codes328

7.4.1 Turbo Decoding of Product Codes330

Problems335

References337

8 Ensemble Enumerators for Turbo and LDPC Codes339

8.1 Notation340

8.2 Ensemble Enumerators for Parallel-Concatenated Codes343

8.2.1 Preliminaries343

8.2.2 PCCC Ensemble Enumerators345

8.3 Ensemble Enumerators for Serial-Concatenated Codes356

8.3.1 Preliminaries356

8.3.2 SCCC Ensemble Enumerators358

8.4 Enumerators for Selected Accumulator-Based Codes362

8.4.1 Enumerators for Repeat-Accumulate Codes362

8.4.2 Enumerators for Irregular Repeat-Accumulate Codes364

8.5 Enumerators for Protograph-Based LDPC Codes367

8.5.1 Finite-Length Ensemble Weight Enumerators368

8.5.2 Asymptotic Ensemble Weight Enumerators371

8.5.3 On the Complexity of Computing Asymptotic Ensemble Enumerators376

8.5.4 Ensemble Trapping-Set Enumerators379

Problems383

References386

9 Ensemble Decoding Thresholds for LDPC and Turbo Codes388

9.1 Density Evolution for Regular LDPC Codes388

9.2 Density Evolution for Irregular LDPC Codes394

9.3 Quantized Density Evolution399

9.4 The Gaussian Approximation402

9.4.1 GA for Regular LDPC Codes403

9.4.2 GA for Irregular LDPC Codes404

9.5 On the Universality of LDPC Codes407

9.6 EXIT Charts for LDPC Codes412

9.6.1 EXIT Charts for Regular LDPC Codes414

9.6.2 EXIT Charts for Irregular LDPC Codes416

9.6.3 EXIT Technique for Protograph-Based Codes417

9.7 EXIT Charts for Turbo Codes420

9.8 The Area Property for EXIT Charts424

9.8.1 Serial-Concatenated Codes424

9.8.2 LDPC Codes425

Problems426

References428

10 Finite-Geometry LDPC Codes430

10.1 Construction of LDPC Codes Based on Lines of Euclidean Geometries430

10.1.1 A Class of Cyclic EG-LDPC Codes432

10.1.2 A Class of Quasi-Cyclic EG-LDPC Codes434

10.2 Construction of LDPC Codes Based on the Parallel Bundles of Lines in Euclidean Geometries436

10.3 Construction of LDPC Codes Based on Decomposition of Euclidean Geometries439

10.4 Construction of EG-LDPC Codes by Masking444

10.4.1 Masking445

10.4.2 Regular Masking446

10.4.3 Irregular Masking447

10.5 Construction of QC-EG-LDPC Codes by Circulant Decomposition450

10.6 Construction of Cyclic and QC-LDPC Codes Based on Projective Geometries455

10.6.1 Cyclic PG-LDPC Codes455

10.6.2 Quasi-Cyclic PG-LDPC Codes458

10.7 One-Step Majority-Logic and Bit-Flipping Decoding Algorithms for FG-LDPC Codes460

10.7.1 The OSMLG Decoding Algorithm for LDPC Codes over the BSC461

10.7.2 The BF Algorithm for Decoding LDPC Codes over the BSC468

10.8 Weighted BF Decoding：Algorithm 1469

10.9 Weighted BF Decoding：Algorithms 2 and 3472

10.10 Concluding Remarks477

Problems477

References481

11 Constructions of LDPC Codes Based on Finite Fields484

11.1 Matrix Dispersions of Elements of a Finite Field484

11.2 A General Construction of QC-LDPC Codes Based on Finite Fields485

11.3 Construction of QC-LDPC Codes Based on the Minimum-Weight Codewords of an RS Code with Two Information Symbols487

11.4 Construction of QC-LDPC Codes Based on the Universal Parity-Check Matrices of a Special Subclass of RS Codes495

11.5 Construction of QC-LDPC Codes Based on Subgroups of a Finite Field501

11.5.1 Construction of QC-LDPC Codes Based on Subgroups of the Additive Group of a Finite Field501

11.5.2 Construction of QC-LDPC Codes Based on Subgroups of the Multiplicative Group of a Finite Field503

11.6 Construction of QC-LDPC Code Based on the Additive Group of a Prime Field506

11.7 Construction of QC-LDPC Codes Based on Primitive Elements of a Field510

11.8 Construction of QC-LDPC Codes Based on the Intersecting Bundles of Lines of Euclidean Geometries512

11.9 A Class of Structured RS-Based LDPC Codes516

Problems520

References522

12 LDPC Codes Based on Combinatorial Designs，Graphs，and Superposition523

12.1 Balanced Incomplete Block Designs and LDPC Codes523

12.2 Class-Ⅰ Bose BIBDs and QC-LDPC Codes524

12.2.1 Class-Ⅰ Bose BIBDs525

12.2.2 Type-Ⅰ Class-Ⅰ Bose BIBD-LDPC Codes525

12.2.3 Type-Ⅱ Class-Ⅰ Bose BIBD-LDPC Codes527

12.3 Class-Ⅱ Bose BIBDs and QC-LDPC Codes530

12.3.1 Class-Ⅱ Bose BIBDs531

12.3.2 Type-Ⅰ Class-Ⅱ Bose BIBD-LDPC Codes531

12.3.3 Type-Ⅱ Class-Ⅱ QC-BIBD-LDPC Codes533

12.4 Construction of Type-Ⅱ Bose BIBD-LDPC Codes by Dispersion536

12.5 A Trellis-Based Construction of LDPC Codes537

12.5.1 A Trellis-Based Method for Removing Short Cycles from a Bipartite Graph538

12.5.2 Code Construction540

12.6 Construction of LDPC Codes Based on Progressive Edge-Growth Tanner Graphs542

12.7 Construction of LDPC Codes by Superposition546

12.7.1 A General Superposition Construction of LDPC Codes546

12.7.2 Construction of Base and Constituent Matrices548

12.7.3 Superposition Construction of Product LDPC Codes552

12.8 Two Classes of LDPC Codes with Girth 8554

Problems557

References559

13 LDPC Codes for Binary Erasure Channels561

13.1 Iterative Decoding of LDPC Codes for the BEC561

13.2 Random-Erasure-Correction Capability563

13.3 Good LDPC Codes for the BEC565

13.4 Correction of Erasure-Bursts570

13.5 Erasure-Burst-Correction Capabilities of Cyclic Finite-Geometry and Superposition LDPC Codes573

13.5.1 Erasure-Burst-Correction with Cyclic Finite-Geometry LDPC Codes573

13.5.2 Erasure-Burst-Correction with Superposition LDPC Codes574

13.6 Asymptotically Optimal Erasure-Burst-Correction QC-LDPC Codes575

13.7 Construction of QC-LDPC Codes by Array Dispersion580

13.8 Cyclic Codes for Correcting Bursts of Erasures586

Problems589

References590

14 Nonbinary LDPC Codes592

14.1 Definitions592

14.2 Decoding of Nonbinary LDPC Codes593

14.2.1 The QSPA593

14.2.2 The FFT-QSPA598

14.3 Construction of Nonbinary LDPC Codes Based on Finite Geometries600

14.3.1 A Class of q m-ary Cyclic EG-LDPC Codes601

14.3.2 A Class of Nonbinary Quasi-Cyclic EG-LDPC Codes607

14.3.3 A Class of Nonbinary Regular EG-LDPC Codes610

14.3.4 Nonbinary LDPC Code Constructions Based on Projective Geometries611

14.4 Constructions of Nonbinary QC-LDPC Codes Based on Finite Fields614

14.4.1 Dispersion of Field Elements into Nonbinary Circulant Permutation Matrices615

14.4.2 Construction of Nonbinary QC-LDPC Codes Based on Finite Fields615

14.4.3 Construction of Nonbinary QC-LDPC Codes by Masking617

14.4.4 Construction of Nonbinary QC-LDPC Codes by Array Dispersion618

14.5 Construction of QC-EG-LDPC Codes Based on Parallel Flats in Euclidean Geometries and Matrix Dispersion620

14.6 Construction of Nonbinary QC-EG-LDPC Codes Based on Intersecting Flats in Euclidean Geometries and Matrix Dispersion624

14.7 Superposition-Dispersion Construction of Nonbinary QC-LDPC Codes628

Problems631

References633

15 LDPC Code Applications and Advanced Topics636

15.1 LDPC-Coded Modulation636

15.1.1 Design Based on EXIT Charts638

15.2 Turbo Equalization and LDPC Code Design for ISI Channels644

15.2.1 Turbo Equalization644

15.2.2 LDPC Code Design for ISI Channels648

15.3 Estimation of LDPC Error Floors651

15.3.1 The Error-Floor Phenomenon and Trapping Sets651

15.3.2 Error-Floor Estimation654

15.4 LDPC Decoder Design for Low Error Floors657

15.4.1 Codes Under Study659

15.4.2 The Bi-Mode Decoder661

15.4.3 Concatenation and Bit-Pinning666

15.4.4 Generalized-LDPC Decoder668

15.4.5 Remarks670

15.5 LDPC Convolutional Codes670

15.6 Fountain Codes672

15.6.1 Tornado Codes673

15.6.2 Luby Transform Codes674

15.6.3 Raptor Codes675

Problems676

References677

Index681