Concise Encyclopedia of Coding Theory

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. Features 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

Part I. Coding Fundamentals 1. Basics of Coding TheoryW. Cary Huffman, Jon-Lark Kim, and Patrick Solé 2. Cyclic Codes over Finite FieldsCunsheng Ding 3. Construction and Classification of Codes.Patric R. J. Östergård 4. Self-Dual CodesStefka Bouyuklieva 5. Codes and DesignsVladimir D. Tonchev 6. Codes over RingsSteven T. Dougherty 7. Quasi-Cyclic CodesCem Güneri, San Ling, and Buket Özkaya 8. Introduction to Skew-Polynomial Rings and Skew-Cyclic CodesHeide Gluesing-Luerssen 9. Additive Cyclic CodesJürgen Bierbrauer, Stefano Marcugini, and Fernanda Pambianco 10. Convolutional CodesJulia Lieb, Raquel Pinto, and Joachim Rosenthal 11. Rank-Metric CodesElisa Gorla 12. Linear Programming BoundsPeter Boyvalenkov and Danyo Danev 13. Semidefinite Programming Bounds for Error-Correcting CodesFrank Vallentin Part II. Families of Codes 14. Coding Theory and Galois GeometriesLeo Storme 15. Algebraic Geometry Codes and Some ApplicationsAlain Couvreur and Hugues Randriambololona 16. Codes in Group AlgebrasWolfgang Willems 17. Constacyclic Codes over Finite Commutative Chain RingsHai Q. Dinh and Sergio R. López-Permouth 18. Weight Distribution of Trace Codes over Finite Rings Minjia Shi 19. Two-Weight CodesAndries E. Brouwer 20. Linear Codes from FunctionsSihem Mesnager 21. Codes over Graphs Christine A. Kelley Part III. Applications 22. Alternative MetricsMarcelo Firer 23. Algorithmic MethodsAlfred Wassermann 24. Interpolation DecodingSwastik Kopparty 25. Pseudo-Noise SequencesTor Helleseth and Chunlei Li 26. Lattice CodingFrédérique Oggier 27. Quantum Error-Control CodesMartianus Frederic Ezerman 28. Space-Time CodingFrédérique Oggier 29. Network CodesFrank R. Kschischang 30. Coding for Erasures and Fountain CodesIan F. Blake 31. Codes for Distributed StorageVinayak Ramkumar, Myna Vajha, S. B. Balaji, M. Nikhil Krishnan, Birenjith Sasidharan, and P. Vijay Kumar 32. Polar CodesNoam Presman and Simon Litsyn 33. Secret Sharing with Linear CodesCunsheng Ding 34. Code-Based CryptographyPhilippe Gaborit and Jean-Christophe Deneuville Bibliography Index

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.