โ Scribed by Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert, Alfred Wassermann (auth.)
No coin nor oath required. For personal study only.
This text offers a thorough introduction to the mathematical concepts behind the theory of error-correcting linear codes. Care is taken to introduce the necessary algebraic concepts, for instance the theory of finite fields, the polynomial rings over such fields and the ubiquitous concept of group actions that allows the classification of codes by isometry. The book provides in-depth coverage of important topics like cyclic codes and the coding theory used in compact disc players.
The final four chapters cover advanced and algorithmic topics like the classification of linear codes by isometry, the enumeration of isometry classes, random generation of codes, the use of lattice basis reduction to compute minimum distances, the explicit construction of codes with given parameters, as well as the systematic evaluation of representatives of all isometry classes of codes. Up until now, these advanced topics have only been covered in research papers.
The present book provides access to these results at a level which is suitable for graduate students of mathematics, computer science and engineering as well as for researchers.
Linear Codes....Pages 1-77
Bounds and Modifications....Pages 79-135
Finite Fields....Pages 137-210
Cyclic Codes....Pages 211-366
Mathematics and Audio Compact Discs....Pages 367-440
Enumeration of Isometry Classes....Pages 441-560
Solving Systems of Diophantine Linear Equations....Pages 561-611
Linear Codes with a Prescribed Minimum Distance....Pages 613-659
The General Case....Pages 661-752
Algebra; Algorithms; Coding and Information Theory; Signal, Image and Speech Processing; Combinatorics
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