Error-Correcting Linear Codes

1. Front Matter

   Pages I-XXIX

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2. Linear Codes

   Pages 1-77

3. Bounds and Modifications

   Pages 79-135

4. Finite Fields

   Pages 137-210

5. Cyclic Codes

   Pages 211-366

6. Mathematics and Audio Compact Discs

   Pages 367-440

7. Enumeration of Isometry Classes

   Pages 441-560

8. Solving Systems of Diophantine Linear Equations

   Pages 561-611

9. Linear Codes with a Prescribed Minimum Distance

   Pages 613-659

10. The General Case

    Pages 661-752

11. Back Matter

    Pages 753-798

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The fascinating theory of error-correcting codes is a rather new addition to the list of mathematical disciplines. It grew out of the need to communicate information electronically, and is currently no more than 60 years old. - ing an applied discipline by de?nition, a surprisingly large number of pure mathematical areas tie into Coding Theory. If one were to name just the most important connections, one would start of course with Linear Algebra, then list Algebra and Combinatorics, and further mention Number Theory and - ometry as well as Algebraic Geometry. Being a thorough introduction to the ?eld, this book starts from the very beginning, which is the channel model of communication in the presence of noise. From there, we develop the fundamental concepts of error-correcting codes, like the Hamming metric and the maximum likelihood decoding pr- ciple. After discussing dual codes and simple decoding procedures, this book takes an unusual turn. The standard approach would be to move on from there and introduce either more theory or present standard constructions of codes. The approach taken here is different.

• Algebra
• Computer
• algorithms
• coding theory
• computer science
• error-correcting codes
• finite field
• linear codes
• combinatorics

From the reviews:

"The theory of error-correcting codes is a … new addition to the list of mathematical disciplines. … This book contains 51 figures and 102 tables. … The book provides access to all results at a level which is proper for graduate students of mathematics and computer science as well as for researchers." (Zlatko Varbanov, Zentralblatt MATH, Vol. 1102 (4), 2007)

"This is a thorough treatment of the theory of error-correcting codes. … This book is remarkable because of the enormous amount of material presented (in a very lucid style), but also because of the great variety of mathematical disciplines used … . A beautiful book on applied mathematics!" (H. Mitsch, Monatshefte für Mathematik, Vol. 151 (3), 2007)

"The main object of the book under review is an error-correcting linear code. … a motivated reader can profit much from studying this monograph, which contains rich material in one of the rapidly developing areas. … The presentation of material is reader-friendly, arguments are clear and concise, numerous exercises are original and stimulating … . To sum up, the book under review can be strongly recommended to anyone interested in the topic." (Boris È. Kunyavskii, Mathematical Reviews, Issue 2008 h)

• Department of Mathematics, Colorado State University, Fort Collins, USA

  Anton Betten

• Siemens CT IC 3, München, Germany

  Michael Braun

• Institut für Mathematik, Karl-Franzens-Universität Graz, Graz, Austria

  Harald Fripertinger

• Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

  Adalbert Kerber, Axel Kohnert, Alfred Wassermann

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