The Theory of Error-Correcting Codes

✍ Scribed by F.J. MacWilliams, N.J.A. Sloane

• Publisher: North Holland Publishing Company
• Year: 1977
• Tongue: English
• Leaves: 785
• Series: North-Holland Mathematical Library 16
• Edition: 1st
• Category: Library

✦ Synopsis

This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings, and Lie groups. The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field. Featured topics- Analysis on manifolds- Differential forms on jet spaces - Global variational functionals- Euler-Lagrange mapping - Helmholtz form and the inverse problem- Symmetries and the Noether's theory of conservation laws- Regularity and the Hamilton theory- Variational sequences - Differential invariants and natural variational principles - First book on the geometric foundations of Lagrange structures- New ideas on global variational functionals - Complete proofs of all theorems - Exact treatment of variational principles in field theory, inc. general relativity- Basic structures and tools: global analysis, smooth manifolds, fibred spaces

✦ Table of Contents

Front Cover......Page 1
Preface......Page 7
Preface to the third printing......Page 13
1. Linear codes......Page 15
2. Nonlinear codes, Hadamardmatrices, designs andthe Golay code......Page 52
3. An introduction to BCH codes and finite fields......Page 94
4. Finite fields......Page 107
5. Dual codes and their weight distribution......Page 139
6. Codes, designs and perfectcodes......Page 169
7. Cyclic codes......Page 202
8. Cyclic codes (cont.): Idempotents and Mattson Solomon polynomials......Page 230
9. BCH codes......Page 271
10. Reed-Solomon and Justesen codes......Page 308
11. MDS codes......Page 331
12. Alternant, Goppa and other generalized BCH codes......Page 346
13. Reed-Muller codes......Page 384
14. First-order Reed-Muller codes......Page 420
15. Second-order Reed-Muller, Kerdock and Preparata codes......Page 447
16. Quadratic-residue codes......Page 494
17. Bounds on the size of a code......Page 537
18. Methods for combining codes......Page 581
19. Self-dual codes and invariant theory......Page 610
20. The Golay codes......Page 648
21. Association schemes......Page 665
Appendix A - Tables of the bestcodes known......Page 687
Appendix B - Finite geometries......Page 706
Bibliography......Page 717
Index......Page 771