Introduction to coding theory

Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, Ron Roth describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This book introduces the reader to the theoretical foundations of this subject with the emphasis on the Reed-Solomon codes and their derivatives. Linear codes -- Introduction to finite fields -- Bounds on the parameters of codes -- Reed-Solomon and related codes -- Decoding of Reed-Solomon codes -- Structure of finite fields -- Cyclic codes -- List decoding of Reed-Solomon codes -- Codes in the Lee metric -- MDS codes -- Concatenated codes -- Graph codes -- Trellis and convolutional codes -- Basics in modern algebra Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This 2006 book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, the author describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail. Homework exercises introduce additional concepts such as Reed-Muller codes, and burst error correction. The end-of-chapter notes often deal with algorithmic issues, such as the time complexity of computational problems. While mathematical rigor is maintained, the text is designed to be accessible to a broad readership, including students of computer science, electrical engineering, and mathematics, from senior-undergraduate to graduate level. Frontmatter......Page 1 Contents......Page 5 Preface......Page 9 1 - Introduction......Page 13 2 - Linear Codes......Page 38 3 - Introduction to Finite Fields......Page 62 4 - Bounds on the Parameters of Codes......Page 105 5 - Reedâ•fiSolomon and Related Codes......Page 159 6 - Decoding of Reedâ•fiSolomon Codes......Page 195 7 - Structure of Finite Fields......Page 230 8 - Cyclic Codes......Page 254 9 - List Decoding of Reedâ•fiSolomon Codes......Page 278 10 - Codes in the Lee Metric......Page 310 11 - MDS Codes......Page 345 12 - Concatenated Codes......Page 377 13 - Graph Codes......Page 407 14 - Trellis and Convolutional Codes......Page 464 Appendix: Basics in Modern Algebra......Page 533 Bibliography......Page 539 List of Symbols......Page 565 Index......Page 571

Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, Ron Roth describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail.

This 2006 book introduces the theoretical foundations of error-correcting codes for senior-undergraduate to graduate students