In recent years quaternary codes have attracted the attention of the coding community as several notorious binary nonlinear codes containing more codewords than any known linear codes were found to be binary images under the Gray map of linear codes over Z 4 This discovery opens the way for a broader study of quaternary codes, which constitute a rapidly growing area of coding theory. This book is based on the author's lectures in Xi'an, China and Lurid, Sweden and covets the most recent developments of this theory.

This book contains survey papers and research papers by leading experts on sequences and their applications. It discusses both the theory of sequences and their applications in cryptography, coding theory, communications systems, numerical computation and computer simulation. Sequences have important applications in ranging systems, spread spectrum communication systems, multi-terminal system identification, code division multiply access communications systems, global positioning systems, software testing, circuit testing, computer simulation, and stream ciphers. The papers contained in this volume bring together experts from discrete mathematics, computer science and communications engineering, and help to bridge advances in these different areas.

This book constitutes the refereed proceedings of the 2nd International Castle Meeting, ISMCTA 2008, Castillo de la Mota, Medina del Campo, Spain, September 2008. The 14 full papers and 5 invited papers presented were carefully reviewed and selected from 34 submissions for inclusion in the book. The papers cover network coding, quantum codes, group codes, codes and combinatorial structures, agebraic-geometry codes, as well as codes and applications.

The theory developed for Z2Z4-additive codes is the starting point for much generalization about codes over mixed alphabets. They have opened a new, emergent area of research. The techniques used for Z2Z4-linear codes are being generalized and applied to more general codes. By example, these codes have contributed to the classification of many nonlinear codes. Moreover, they can be considered as the starting point of many different generalizations given over mixed alphabets, thereby representing a useful area of research. Since 2010, more than 30 papers have been published about the codes considered in the book, which includes important classes of binary codes (1-perfect, Hadamard, etc.) that are not linear in general. For example, much recent research has shown the application of the techniques described for the family of cyclic Z2Z4-linear codes. Topics and Features: Examines everything from the basic definitions to very advanced results Provides numerous examples, succinctly and comprehensively gathering and using the relevant information Includes examples using commands from a new Magma package, developed mostly by the same authors Proposes algorithms, for instance to describe coding and decoding strategies This unique volume will be eminently suitable for researchers working on coding theory over rings, especially over mixed alphabets. Experts will find commands and algorithms that will be useful in the generalization to codes over mixed alphabets. Additionally, by outlining the basic theory of codes over mixed alphabets and providing numerous examples, the book will be useful to researchers wanting to be introduced to the topic. The authors are all affiliated with the Dept. of Information and Communications Engineering at the Universitat Autònoma de Barcelona, Spain. Joaquim Borges and Cristina Fernández-Córdoba are Associate Professors, Jaume Pujol is a now retired Associate Professor, Josep Rifà is Professor Emeritus, and Mercè Villanueva is Associate Professor.

This book is designed to be usable as a textbook for an undergraduate course or for an advanced graduate course in coding theory as well as a reference for researchers in discrete mathematics, engineering and theoretical computer science. This second edition has three parts: an elementary introduction to coding, theory and applications of codes, and algebraic curves. The latter part presents a brief introduction to the theory of algebraic curves and its most important applications to 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

It is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10.