Algebraic Codes for Data Transmission

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ISBN: 9780521553742 | 0521553741
Cover: Hardcover
Copyright: 3/10/2003

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The need to transmit and store massive amounts of data reliably and without error is a vital part of modern communications systems. Error-correcting codes play a fundamental role in minimizing data corruption caused by defects such as noise, interference, crosstalk and packet loss. This book provides an accessible introduction to the basic elements of algebraic codes, and discusses their use in a variety of applications. The author describes a range of important coding techniques, including Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. Throughout the book, mathematical theory is illustrated by reference to many practical examples. The book is aimed at graduate students of electrical and computer engineering, and at practising engineers whose work involves communications or signal processing.

Preface

1 Introduction

(19)

1.1 The discrete communication channel

(2)

1.2 The history of data-transmission codes

(2)

1.3 Applications

(1)

1.4 Elementary concepts

(7)

1.5 Elementary codes

(3)

Problems

(3)

2 Introduction to Algebra

(29)

2.1 Fields of characteristic two

(3)

2.2 Groups

(5)

2.3 Rings

(2)

2.4 Fields

(2)

2.5 Vector spaces

(5)

2.6 Linear algebra

(8)

Problems

(3)

Notes

(1)

3 Linear Block Codes

(18)

3.1 Structure of linear block codes

(1)

3.2 Matrix description of linear block codes

(4)

3.3 Hamming codes

(2)

3.4 The standard array

(3)

3.5 Hamming spheres and perfect codes

(3)

3.6 Simple modifications to a linear code

(1)

Problems

(3)

Notes

(1)

4 The Arithmetic of Galois Fields

(29)

4.1 The integer ring

(3)

4.2 Finite fields based on the integer ring

(2)

4.3 Polynomial rings

(7)

4.4 Finite fields based on polynomial rings

(4)

4.5 Primitive elements

(3)

4.6 The structure of finite fields

(6)

Problems

(3)

Notes

(1)

5 Cyclic Codes

(35)

5.1 Viewing a code from an extension field

(3)

5.2 Polynomial description of cyclic codes

(5)

5.3 Minimal polynomials and conjugates

104

(7)

5.4 Matrix description of cyclic codes

111

(2)

5.5 Hamming codes as cyclic codes

113

(3)

5.6 Cyclic codes for correcting double errors

116

(2)

5.7 Quasi-cyclic codes and shortened cyclic codes

118

(1)

5.8 The Golay code as a cyclic code

119

(4)

5.9 Cyclic codes for correcting burst errors

123

(2)

5.10 The Fire codes as cyclic codes

125

(2)

5.11 Cyclic codes for error detection

127

(1)

Problems

128

(2)

Notes

130

(1)

6 Codes Based on the Fourier Transform

131

(48)

6.1 The Fourier transform

131

(7)

6.2 Reed-Solomon codes

138

(5)

6.3 Conjugacy constraints and idempotents

143

(5)

6.4 Spectral description of cyclic codes

148

(4)

6.5 BCH codes

152

(7)

6.6 The Peterson-Gorenstein-Zierler decoder

159

(7)

6.7 The Reed-Muller codes as cyclic codes

166

(3)

6.8 Extended Reed-Solomon codes

169

(3)

6.9 Extended BCH codes

172

(3)

Problems

175

(2)

Notes

177

(2)

7 Algorithms Based on the Fourier Transform

179

(49)

7.1 Spectral estimation in a finite field

179

(4)

7.2 Synthesis of linear recursions

183

(8)

7.3 Decoding of binary BCH codes

191

(2)

7.4 Decoding of nonbinary BCH codes

193

(8)

7.5 Decoding with erasures and errors

201

(5)

7.6 Decoding in the time domain

206

(4)

7.7 Decoding within the BCH bound

210

(3)

7.8 Decoding beyond the BCH bound

213

(3)

7.9 Decoding of extended Reed-Solomon codes

216

(1)

7.10 Decoding with the euclidean algorithm

217

(6)

Problems

223

(3)

Notes

226

(2)

8 Implementation

228

(42)

8.1 Logic circuits for finite-field arithmetic

228

(7)

8.2 Shift-register encoders and decoders

235

(2)

8.3 The Meggitt decoder

237

(7)

8.4 Error trapping

244

(6)

8.5 Modified error trapping

250

(4)

8.6 Architecture of Reed-Solomon decoders

254

(4)

8.7 Multipliers and inverters

258

(4)

8.8 Bit-serial multipliers

262

(5)

Problems

267

(2)

Notes

269

(1)

9 Convolutional Codes

270

(43)

9.1 Codes without a block structure

270

(3)

9.2 Trellis description of convolutional codes

273

(5)

9.3 Polynomial description of convolutional codes

278

(4)

9.4 Check matrices and inverse matrices

282

(5)

9.5 Error correction and distance notions

287

(2)

9.6 Matrix description of convolutional codes

289

(2)

9.7 The Wyner-Ash codes as convolutional codes

291

(3)

9.8 Syndrome decoding algorithms

294

(4)

9.9 Convolutional codes for correcting error bursts

298

(5)

9.10 Algebraic structure of convolutional codes

303

(6)

Problems

309

(2)

Notes

311

(2)

10 Beyond BCH Codes

313

(22)

10.1 Product codes and interleaved codes

314

(4)

10.2 Bicyclic codes

318

(3)

10.3 Concatenated codes

321

(2)

10.4 Cross-interleaved codes

323

(3)

10.5 Turbo codes

326

(3)

10.6 Justesen codes

329

(3)

Problems

332

(2)

Notes

334

(1)

11 Codes and Algorithms Based on Graphs

335

(40)

11.1 Distance, probability, and likelihood

336

(4)

11.2 The Viterbi algorithm

340

(3)

11.3 Sequential algorithms to search a trellis

343

(7)

11.4 Trellis description of linear block codes

350

(4)

11.5 Gallager codes

354

(1)

11.6 Tanner graphs and factor graphs

355

(2)

11.7 Posterior probabilities

357

(2)

11.8 The two-way algorithm

359

(3)

11.9 Iterative decoding of turbo codes

362

(2)

11.10 Tail-biting representations of block codes

364

(4)

11.11 The Golay code as a tail-biting code

368

(4)

Problems

372

(2)

Notes

374

(1)

12 Performance of Error-Control Codes

375

(43)

12.1 Weight distributions of block codes

375

(8)

12.2 Performance of block codes

383

(3)

12.3 Bounds on minimum distance of block codes

386

(8)

12.4 Binary expansions of Reed-Solomon codes

394

(5)

12.5 Symbol error rates on a gaussian-noise channel

399

(4)

12.6 Sequence error rates on a gaussian-noise channel

403

(3)

12.7 Coding gain

406

(5)

12.8 Capacity of a gaussian-noise channel

411

(3)

Problems

414

(2)

Notes

416

(2)

13 Codes and Algorithms for Majority Decoding

418

(45)

13.1 Reed-Muller codes

418

(8)

13.2 Decoding by majority vote

426

(4)

13.3 Circuits for majority decoding

430

(3)

13.4 Affine permutations for cyclic codes

433

(4)

13.5 Cyclic codes based on permutations

437

(4)

13.6 Convolutional codes for majority decoding

441

(1)

13.7 Generalized Reed-Muller codes

442

(5)

13.8 Euclidean-geometry codes

447

(9)

13.9 Projective-geometry codes

456

(4)

Problems

460

(1)

Notes

461

(2)

Bibliography

463

(10)

Index

473

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Algebraic Codes for Data Transmission

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Table of Contents

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