《Introduction to the theory of error-correcting codes Part II》

Chapter 1. Linear codes1

Chapter 2. Nonlinear codes，Hadamard matrices，designs and the Golay code38

Chapter 3. An introduction to BCH codes and finite fields80

Chapter 4. Finite fields93

Chapter 5. Dual codes and their weight distribution125

Chapter 6. Codes，designs and perfect codes155

Chapter 7. Cyclic codes188

Chapter 8. Cyclic codes （contd.）： Idempotents and Mattson-Solomon polynomials216

Chapter 9. BCH codes257

Chapter 10. Reed-Solomon and Justesen codes294

Chapter 11. MDS codes317

Chapter 12. Alternant，Goppa and other generalized BCH codes332

Chapter 13.Reed-Muller codes370

1.Introduction370

2.Boolean functions370

3.Reed-Muller Codes373

4.RM codes and geometries377

5.The minimum weight vectors generate the code381

6.Encoding and decoding （Ⅰ）385

7.Encoding and decoding （Ⅱ）388

8.Other geometrical codes397

9.Automorphism groups of the RM codes398

10.Mattson-Solomon polynomials of RM codes401

11.The action of the general affine group on Mattson-Solomon polynomials402

Notes on Chapter 13403

Chapter 14.First-order Reed-Muller codes406

1.Introduction406

2.Pseudo-noise sequences406

3.Cosets of the first-order Reed-Muller code412

4.Encoding and decoding ？ （l，m）419

5.Bent functions426

Notes on Chapter 14431

Chapter 15.Second-order Reed-Muller，Kerdock and Preparata codes433

1.Introduction433

2.Weight distribution of second-order Reed-Muller codes434

3.Weight distribution of arbitrary Reed-Muller codes445

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

5.The Kerdock code and generalizations453

6.The Preparata code466

7.Goethals’ generalization of the Preparata codes476

Notes on Chapter 15477

Chapter 16.Quadratic-residue codes480

1.Introduction480

2.Definition of quadratic-residue codes481

3.Idempotents of quadratic-residue codes484

4.Extended quadratic-residue codes488

5.The automorphism group of QR codes491

6.Binary quadratic residue codes494

7.Double circulant and quasi-cyclic codes505

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

9.Decoding of cyclic codes and others512

Notes on Chapter 16518

Chapter 17.Bounds on the size of a code523

1.Introduction523

2.Bounds on A（n，d，w）524

3.Bounds on A（n，d）531

4.Linear programming bounds535

5.The Griesmer bound546

6.Constructing linear codes； anticodes547

7.Asymptotic bounds556

Notes on Chapter 17566

Chapter 18.Methods for combining codes567

1.Introduction567

Part Ⅰ： Product codes and generalizations568

2.Direct product codes568

3.Not all cyclic codes are direct products of cyclic codes571

4.Another way of factoring irreducible cyclic codes573

5.Concatenated codes： the ＊ construction575

6.A general decomposition for cyclic codes578

Part Ⅱ： Other methods of combining codes581

7.Methods which increase the length581

7.1 Construction X： adding tails to the codewords581

7.2 Construction X4： combining four codes584

7.3 Single- and double-error-correcting codes586

7.4 The |a ＋ x|b ＋ x|a ＋ b ＋ x| construction587

7.5 Piret’s construction588

8.Constructions related to concatenated codes589

8.1 A method for improving concatenated codes589

8.2 Zinov’ev’s generalized concatenated codes590

9.Methods for shortening a code592

9.1 Constructions Y 1-Y4592

9.2 A construction of Helgert and Stinaff593

Notes on Chapter 18594

Chapter 19.Self-dual codes and invariant theory596

1.Introduction596

2.An introduction to invariant theory598

3.The basic theorems of invariant theory607

4.Generalizations of Gleason’s theorems617

5.The nonexistence of certain very good codes624

6.Good self-dual codes exist629

Notes on Chapter 19633

Chapter 20.The Golay codes634

1.Introduction634

2.The Mathieu group M24636

3.M24 is five-fold transitive637

4.The order of M24 is 24.23.22.21.20.48638

5.The Steiner system S（5，8，24） is unique641

6.The Golay codes ？23 and ？24 are unique646

7.The automorphism groups of the ternary Golay codes647

8.The Golay codes ？11 and ？12 are unique648

Notes on Chapter 20649

Chapter 21.Association schemes651

1.Introduction651

2.Association schemes651

3.The Hamming association scheme656

4.Metric schemes659

5.Symplectic forms661

6.The Johnson scheme665

7.Subsets of association schemes666

8.Subsets of symplectic forms667

9.t-designs and orthogonal arrays670

Notes on Chapter 21671

Appendix A.Tables of the best codes known673

1.Introduction673

2.Figure 1，a small table of A（n，d）683

3.Figure 2，an extended table of the best codes known690

4.Figure 3，a table of A（n，d，w）691

Appendix B.Finite geometries692

1.Introduction692

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 planes701

Notes on Appendix B702

Bibliography703

Index757