Introduction to the theory of error-correcting codes Part IIPDF电子书下载

Chapter 1. Linear codes 1

Chapter 2. Nonlinear codes，Hadamard matrices，designs and the Golay code 38

Chapter 3. An introduction to BCH codes and finite fields 80

Chapter 4. Finite fields 93

Chapter 5. Dual codes and their weight distribution 125

Chapter 6. Codes，designs and perfect codes 155

Chapter 7. Cyclic codes 188

Chapter 8. Cyclic codes （contd.）： Idempotents and Mattson-Solomon polynomials 216

Chapter 9. BCH codes 257

Chapter 10. Reed-Solomon and Justesen codes 294

Chapter 11. MDS codes 317

Chapter 12. Alternant，Goppa and other generalized BCH codes 332

Chapter 13.Reed-Muller codes 370

1.Introduction 370

2.Boolean functions 370

3.Reed-Muller Codes 373

4.RM codes and geometries 377

5.The minimum weight vectors generate the code 381

6.Encoding and decoding （Ⅰ） 385

7.Encoding and decoding （Ⅱ） 388

8.Other geometrical codes 397

9.Automorphism groups of the RM codes 398

10.Mattson-Solomon polynomials of RM codes 401

11.The action of the general affine group on Mattson-Solomon polynomials 402

Notes on Chapter 13 403

Chapter 14.First-order Reed-Muller codes 406

1.Introduction 406

2.Pseudo-noise sequences 406

3.Cosets of the first-order Reed-Muller code 412

4.Encoding and decoding ？ （l，m） 419

5.Bent functions 426

Notes on Chapter 14 431

Chapter 15.Second-order Reed-Muller，Kerdock and Preparata codes 433

1.Introduction 433

2.Weight distribution of second-order Reed-Muller codes 434

3.Weight distribution of arbitrary Reed-Muller codes 445

4.Subcodes of dimension 2m of ？（2，m）＊ and ?（2，m） 448

5.The Kerdock code and generalizations 453

6.The Preparata code 466

7.Goethals’ generalization of the Preparata codes 476

Notes on Chapter 15 477

Chapter 16.Quadratic-residue codes 480

1.Introduction 480

2.Definition of quadratic-residue codes 481

3.Idempotents of quadratic-residue codes 484

4.Extended quadratic-residue codes 488

5.The automorphism group of QR codes 491

6.Binary quadratic residue codes 494

7.Double circulant and quasi-cyclic codes 505

8.Quadratic-residue and symmetry codes over GF（3） 510

9.Decoding of cyclic codes and others 512

Notes on Chapter 16 518

Chapter 17.Bounds on the size of a code 523

1.Introduction 523

2.Bounds on A（n，d，w） 524

3.Bounds on A（n，d） 531

4.Linear programming bounds 535

5.The Griesmer bound 546

6.Constructing linear codes； anticodes 547

7.Asymptotic bounds 556

Notes on Chapter 17 566

Chapter 18.Methods for combining codes 567

1.Introduction 567

Part Ⅰ： Product codes and generalizations 568

2.Direct product codes 568

3.Not all cyclic codes are direct products of cyclic codes 571

4.Another way of factoring irreducible cyclic codes 573

5.Concatenated codes： the ＊ construction 575

6.A general decomposition for cyclic codes 578

Part Ⅱ： Other methods of combining codes 581

7.Methods which increase the length 581

7.1 Construction X： adding tails to the codewords 581

7.2 Construction X4： combining four codes 584

7.3 Single- and double-error-correcting codes 586

7.4 The |a ＋ x|b ＋ x|a ＋ b ＋ x| construction 587

7.5 Piret’s construction 588

8.Constructions related to concatenated codes 589

8.1 A method for improving concatenated codes 589

8.2 Zinov’ev’s generalized concatenated codes 590

9.Methods for shortening a code 592

9.1 Constructions Y 1-Y4 592

9.2 A construction of Helgert and Stinaff 593

Notes on Chapter 18 594

Chapter 19.Self-dual codes and invariant theory 596

1.Introduction 596

2.An introduction to invariant theory 598

3.The basic theorems of invariant theory 607

4.Generalizations of Gleason’s theorems 617

5.The nonexistence of certain very good codes 624

6.Good self-dual codes exist 629

Notes on Chapter 19 633

Chapter 20.The Golay codes 634

1.Introduction 634

2.The Mathieu group M24 636

3.M24 is five-fold transitive 637

4.The order of M24 is 24.23.22.21.20.48 638

5.The Steiner system S（5，8，24） is unique 641

6.The Golay codes ？23 and ？24 are unique 646

7.The automorphism groups of the ternary Golay codes 647

8.The Golay codes ？11 and ？12 are unique 648

Notes on Chapter 20 649

Chapter 21.Association schemes 651

1.Introduction 651

2.Association schemes 651

3.The Hamming association scheme 656

4.Metric schemes 659

5.Symplectic forms 661

6.The Johnson scheme 665

7.Subsets of association schemes 666

8.Subsets of symplectic forms 667

9.t-designs and orthogonal arrays 670

Notes on Chapter 21 671

Appendix A.Tables of the best codes known 673

1.Introduction 673

2.Figure 1，a small table of A（n，d） 683

3.Figure 2，an extended table of the best codes known 690

4.Figure 3，a table of A（n，d，w） 691

Appendix B.Finite geometries 692

1.Introduction 692

2.Finite geometries，PG（m，q） and EG（m，q） 692

3.Properties of PG（m，q） and EG（m，q） 697

4.Projective and affine planes 701

Notes on Appendix B 702

Bibliography 703

Index 757