Most coding theory experts date the origin of the subject with the 1948 publication of A Mathematical Theory of Communication by Claude Shannon. Since then, coding theory has grown into a discipline with many practical applications (antennas, networks, memories), requiring various mathematical techniques, from commutative algebra, to semi-definite programming, to algebraic geometry.
Most topics covered in the Concise Encyclopedia of Coding Theory are presented in short sections at an introductory level and progress from basic to advanced level, with definitions, examples, and many references.
The book is divided into three parts:
Part I fundamentals: cyclic codes, skew cyclic codes, quasi-cyclic codes, self-dual codes, codes and designs, codes over rings, convolutional codes, performance bounds
Part II families: AG codes, group algebra codes, few-weight codes, Boolean function codes, codes over graphs
Part III applications: alternative metrics, algorithmic techniques, interpolation decoding, pseudo-random sequences, lattices, quantum coding, space-time codes, network coding, distributed storage, secret-sharing, and code-based-cryptography.
- Suitable for students and researchers in a wide range of mathematical disciplines
- Contains many examples and references
- Most topics take the reader to the frontiers of research
Table of Contents
Part I. Coding Fundamentals
1. Basics of Coding Theory
W. Cary Huffman, Jon-Lark Kim, and Patrick Solé
2. Cyclic Codes over Finite Fields
3. Construction and Classification of Codes.
Patric R. J. Östergård
4. Self-Dual Codes
5. Codes and Designs
Vladimir D. Tonchev
6. Codes over Rings
Steven T. Dougherty
7. Quasi-Cyclic Codes
Cem Güneri, San Ling, and Buket Özkaya
8. Introduction to Skew-Polynomial Rings and Skew-Cyclic Codes
9. Additive Cyclic Codes
Jürgen Bierbrauer, Stefano Marcugini, and Fernanda Pambianco
10. Convolutional Codes
Julia Lieb, Raquel Pinto, and Joachim Rosenthal
11. Rank-Metric Codes
12. Linear Programming Bounds
Peter Boyvalenkov and Danyo Danev
13. Semidefinite Programming Bounds for Error-Correcting Codes
Part II. Families of Codes
14. Coding Theory and Galois Geometries
15. Algebraic Geometry Codes and Some Applications
Alain Couvreur and Hugues Randriambololona
16. Codes in Group Algebras
17. Constacyclic Codes over Finite Commutative Chain Rings
Hai Q. Dinh and Sergio R. López-Permouth
18. Weight Distribution of Trace Codes over Finite Rings
19. Two-Weight Codes
Andries E. Brouwer
20. Linear Codes from Functions
21. Codes over Graphs
Christine A. Kelley
Part III. Applications
22. Alternative Metrics
23. Algorithmic Methods
24. Interpolation Decoding
25. Pseudo-Noise Sequences
Tor Helleseth and Chunlei Li
26. Lattice Coding
27. Quantum Error-Control Codes
Martianus Frederic Ezerman
28. Space-Time Coding
29. Network Codes
Frank R. Kschischang
30. Coding for Erasures and Fountain Codes
Ian F. Blake
31. Codes for Distributed Storage
Vinayak Ramkumar, Myna Vajha, S. B. Balaji, M. Nikhil Krishnan, Birenjith Sasidharan, and P. Vijay Kumar
32. Polar Codes
Noam Presman and Simon Litsyn
33. Secret Sharing with Linear Codes
34. Code-Based Cryptography
Philippe Gaborit and Jean-Christophe Deneuville
W. Cary Huffman (1948-) received his PhD in mathematics from the California Institute of Technology in 1974. He taught at Dartmouth College (1974 – 1976) as a John Wesley Young Research Instructor and then at Union College (1976 – 1978). In 1978, he joined the Department of Mathematics and Statistics at Loyola University Chicago, continuing there until his retirement in 2018; he is now professor emeritus. He served as that department’s chair from 1986 to 1992. He is co-editor of the Handbook of Coding Theory and co-author of Fundamentals of Error-Correcting Codes, both with Vera Pless. In addition, he has published numerous papers in finite group theory, combinatorics, and algebraic coding theory.
Jon-Lark Kim received his Ph.D. in 2002 from Department of Math of the University of Illinois at Chicago. He was an Associate Professor at the University of Louisville until 2012. He is currently professor at Math Department of Sogang University in Seoul. He has authored more than fifty research papers and one book on Coding Theory. He is the recipient of the 2004 Kirkman medal from the Institute of Combinatorics and its Applications. His research interests include Coding Theory, Cryptography, Combinatorics, Bioinformatics, and Artificial Intelligence.
Patrick Solé (1960-) received the Ingénieur and the Docteur Ingénieur degrees from Ecole Nationale Supérieure des Télécommunications, Paris, France in 1984 and 1987, respectively, and the Habilitation à Diriger des Recherches degree from University of Nice Sophia-Antipolis, France, in 1993.
He has held several visiting positions at Syracuse University, Syracuse, NY, in 1987 – 1989, Macquarie University, Sydney, Australia, in 1994 – 1996, and at University des Sciences et Techniques de Lille, Lille, France, in 1999 – 2000. He has been a permanent member of Centre National de la Recherche Scientifique since 1989 and was later promoted to the rank of Senior Researcher (Directeur de Recherche) in 1996.
His research interests include coding theory (covering radius, codes over rings, geometric codes, quantum codes), interconnection networks (graph spectra, expanders), space time codes (lattices, theta series), and cryptography (Boolean functions, secret sharing schemes).
He is the author of more than 200 journal papers, and of four books.
"If you want a comprehensive outline of classical coding theory and its modern applications, you will appreciate this encyclopedia. It is written by a large international group of experts with a wide spread of experience. Extensive references to publications and web sources support the chapters; they allow the reader to follow in depth interests sparked by each topic covered. The vast range of applications detailed makes it evident that coding theory is a major and vital area of mathematics.""Featuring a wealth of examples and references, and addressing topics on the cutting edge of research and practice, Concise Encyclopedia of Coding Theory is especially useful for students and researchers in a wide range of mathematical disciplines, as well as is an ideal textbook for coding theory curriculums. While especially and unreservedly recommended for college and university library Combinatorics, Information Theory, and Web Encryption collections and supplemental studies curriculums, it should be noted for the personal reading lists of professional coders, coding students, academia, and non-specialist general readers with an interest in the subject"
– Emeritus Professor Harold Ward, University of Virginia
– Midwest Book Review