Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Linear Codes
- 3 Introduction to Finite Fields
- 4 Bounds on the Parameters of Codes
- 5 Reed–Solomon and Related Codes
- 6 Decoding of Reed–Solomon Codes
- 7 Structure of Finite Fields
- 8 Cyclic Codes
- 9 List Decoding of Reed–Solomon Codes
- 10 Codes in the Lee Metric
- 11 MDS Codes
- 12 Concatenated Codes
- 13 Graph Codes
- 14 Trellis and Convolutional Codes
- Appendix: Basics in Modern Algebra
- Bibliography
- List of Symbols
- Index
1 - Introduction
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Linear Codes
- 3 Introduction to Finite Fields
- 4 Bounds on the Parameters of Codes
- 5 Reed–Solomon and Related Codes
- 6 Decoding of Reed–Solomon Codes
- 7 Structure of Finite Fields
- 8 Cyclic Codes
- 9 List Decoding of Reed–Solomon Codes
- 10 Codes in the Lee Metric
- 11 MDS Codes
- 12 Concatenated Codes
- 13 Graph Codes
- 14 Trellis and Convolutional Codes
- Appendix: Basics in Modern Algebra
- Bibliography
- List of Symbols
- Index
Summary
In this chapter, we introduce the model of a communication system, as originally proposed by Claude E. Shannon in 1948. We will then focus on the channel portion of the system and define the concept of a probabilistic channel, along with models of an encoder and a decoder for the channel. As our primary example of a probabilistic channel—here, as well as in subsequent chapters—we will introduce the memoryless q-ary symmetric channel, with the binary case as the prevailing instance used in many practical applications. For q = 2 (the binary case), we quote two key results in information theory. The first result is a coding theorem, which states that information through the channel can be transmitted with an arbitrarily small probability of decoding error, as long as the transmission rate is below a quantity referred to as the capacity of the channel. The second result is a converse coding theorem, which states that operating at rates above the capacity necessarily implies unreliable transmission.
In the remaining part of the chapter, we shift to a combinatorial setting and characterize error events that can occur in channels such as the q-ary symmetric channel, and can always be corrected by suitably selected encoders and decoders. We exhibit the trade-off between error correction and error detection: while an error-detecting decoder provides less information to the receiver, it allows us to handle twice as many errors.
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- Information
- Introduction to Coding Theory , pp. 1 - 25Publisher: Cambridge University PressPrint publication year: 2006