Z2Z4-Linear Codes

The theory developed for 2 4-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 2 4-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 2 4-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.

  There is no other book of Z2Z4-linear code It is the start point of mixed alphabet codes The book contains a huge number of examples, some of them based in the Magma software

  Autorentext

  Joaquim Borges was born in Catalonia in October 1965. He received the B.Sc. degree in Sciences in 1988 from the Universitat Autònoma de Barcelona (UAB) and the Ph.D. degree in Computer Science Engineering in 1998 from the same university. Since 1988, he has been with the Dept. of Information and Communications Engineering at the UAB, where he is currently Associate Professor. His research interests include subjects related to combinatorics, coding theory, and graph theory.

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  Cristina Fernández-Córdoba was born in Catalonia in 1977. She received her B.Sc. degree in Mathematics in 2000, and the Ph.D. degree in Computer Science in 2005 from the UAB. In 2000, she joined the Dpt. of Information and Communications Engineering (dEIC) at the UAB until 2008, when she joined the FECYT with a Fulbright grant. Since 2009, she has been at dEIC, where she is currently an Associate Professor.

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  Jaume Pujol was born in Catalonia in 1955. He received the B.Sc. degree in Mathematics in 1978, the B.Sc. degree in Computer Sciences in 1989 and the Ph.D. degree in Computer Science in 1995 from the UAB. Since 1988 he has been with the Dept. of Information and Communications Engineering at the UAB, as Assistant Professor, and was promoted to Associate Professor in 1996, until his retirement in 2016. His research interests include subjects related to combinatorics, algebra, coding theory, and graph theory.

  Josep Rifà was born in Catalonia in July 1951. He received the Ph.D. in Computer Sciences in 1987 and since 1992 he has been a full professor at the UAB. He was the former Head of the Dpt. of Information and Communications Engineering at UAB as well as the former Vice-chairman of the Spanish Chapter of IT-IEEE. He has mainly worked in several projects of Spanish National R&D&I plan related to digital communications, error correcting codes, and cryptography.

  Mercè Villanueva was born in Catalonia in 1972. She received the B.Sc. degree in Mathematics in 1994, the M.Sc. degree and Ph.D. degree in Computer Science in 1996 and 2001, respectively, from the UAB. In 1994, she joined the Dpt. of Information and Communications Engineering at the UAB, as an Assistant Professor, and was promoted to Associate Professor in 2002. Her research interests include subjects related to combinatorics, algebra, coding theory, and graph theory.

  Klappentext

  The theory developed for 2 4-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 2 4-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 2 4-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 areAssociate Professors, Jaume Pujol is a now retired Associate Professor, Josep Rifà is Professor Emeritus, and Mercè Villanueva is Associate Professor.

Inhalt

1 Introduction.- 2 2 4-Additive and 2 4-Linear Codes.- 3 Duality of 2 4-Additive Codes.- 4 2 4-Additive Self-Dual Codes.- 5 Linearity, Rank and Kernel.- 6 Families of 2 4-Additive Codes.- 7 2 4-Additive Cyclic Codes.- 8 Encoding and Decoding 2 4-Linear Codes.- 9 Generalizations and Applications of 2 4-Additive Codes.