Book contents
- Frontmatter
- Contents
- Preface
- Dedication
- 1 Introduction
- 2 Introduction to Algebra
- 3 Linear Block Codes
- 4 The Arithmetic of Galois Fields
- 5 Cyclic Codes
- 6 Codes Based on the Fourier Transform
- 7 Algorithms Based on the Fourier Transform
- 8 Implementation
- 9 Convolutional Codes
- 10 Beyond BCH Codes
- 11 Codes and Algorithms Based on Graphs
- 12 Performance of Error-Control Codes
- 13 Codes and Algorithms for Majority Decoding
- Bibliography
- Index
5 - Cyclic Codes
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Dedication
- 1 Introduction
- 2 Introduction to Algebra
- 3 Linear Block Codes
- 4 The Arithmetic of Galois Fields
- 5 Cyclic Codes
- 6 Codes Based on the Fourier Transform
- 7 Algorithms Based on the Fourier Transform
- 8 Implementation
- 9 Convolutional Codes
- 10 Beyond BCH Codes
- 11 Codes and Algorithms Based on Graphs
- 12 Performance of Error-Control Codes
- 13 Codes and Algorithms for Majority Decoding
- Bibliography
- Index
Summary
Cyclic codes are a subclass of the class of linear codes obtained by imposing an additional strong structural requirement on the codes. Because of this structure, the search for error-control codes has been most successful within the class of cyclic codes. Here the theory of Galois fields has been used as a mathematical searchlight to spot the good codes. Outside the class of cyclic codes, the theory of Galois fields casts a dimmer light. Most of what has been accomplished builds on the ideas developed for cyclic codes.
The cyclic property in itself is not important and cyclic codes do not necessarily have a large minimum distance. Cyclic codes are introduced because their structure is closely related to the strong structure of the Galois fields. This is significant because the underlying Galois-field description of a cyclic code leads to encoding and decoding procedures that are algorithmic and computationally efficient. Algorithmic techniques have important practical applications, in contrast to the tabular decoding techniques that are used for arbitrary linear codes.
This chapter gives a leisurely introduction to cyclic codes as a special class of linear codes. An alternative approach to the topic of cyclic codes, based on the Fourier transform, is given in Chapter 6.
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- Chapter
- Information
- Algebraic Codes for Data Transmission , pp. 96 - 130Publisher: Cambridge University PressPrint publication year: 2003