Error-Correcting Linear Codes [electronic resource] : Classification by Isometry and Applications / by Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert, Alfred Wassermann.
Contributor(s): Braun, Michael [author.] | Fripertinger, Harald [author.] | Kerber, Adalbert [author.] | Kohnert, Axel [author.] | Wassermann, Alfred [author.] | SpringerLink (Online service)Material type: TextSeries: Algorithms and Computation in Mathematics: 18Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2006Description: XXIX, 798 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783540317036Subject(s): Mathematics | Coding theory | Algebra | Algorithms | Combinatorics | Mathematics | Algebra | Algorithms | Coding and Information Theory | Signal, Image and Speech Processing | CombinatoricsAdditional physical formats: Printed edition:: No titleDDC classification: 512 LOC classification: QA150-272Online resources: Click here to access online
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Linear Codes -- Bounds and Modifications -- Finite Fields -- Cyclic Codes -- Mathematics and Audio Compact Discs -- Enumeration of Isometry Classes -- Solving Systems of Diophantine Linear Equations -- Linear Codes with a Prescribed Minimum Distance -- The General Case.
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.