Book contents
- Frontmatter
- Contents
- Preface
- 1 Basic concepts of linear codes
- 2 Bounds on the size of codes
- 3 Finite fields
- 4 Cyclic codes
- 5 BCH and Reed–Solomon codes
- 6 Duadic codes
- 7 Weight distributions
- 8 Designs
- 9 Self-dual codes
- 10 Some favorite self-dual codes
- 11 Covering radius and cosets
- 12 Codes over ℤ4
- 13 Codes from algebraic geometry
- 14 Convolutional codes
- 15 Soft decision and iterative decoding
- References
- Symbol index
- Subject index
1 - Basic concepts of linear codes
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Basic concepts of linear codes
- 2 Bounds on the size of codes
- 3 Finite fields
- 4 Cyclic codes
- 5 BCH and Reed–Solomon codes
- 6 Duadic codes
- 7 Weight distributions
- 8 Designs
- 9 Self-dual codes
- 10 Some favorite self-dual codes
- 11 Covering radius and cosets
- 12 Codes over ℤ4
- 13 Codes from algebraic geometry
- 14 Convolutional codes
- 15 Soft decision and iterative decoding
- References
- Symbol index
- Subject index
Summary
In 1948 Claude Shannon published a landmark paper “A mathematical theory of communication” that signified the beginning of both information theory and coding theory. Given a communication channel which may corrupt information sent over it, Shannon identified a number called the capacity of the channel and proved that arbitrarily reliable communication is possible at any rate below the channel capacity. For example, when transmitting images of planets from deep space, it is impractical to retransmit the images. Hence if portions of the data giving the images are altered, due to noise arising in the transmission, the data may prove useless. Shannon's results guarantee that the data can be encoded before transmission so that the altered data can be decoded to the specified degree of accuracy. Examples of other communication channels include magnetic storage devices, compact discs, and any kind of electronic communication device such as cellular telephones.
The common feature of communication channels is that information is emanating from a source and is sent over the channel to a receiver at the other end. For instance in deep space communication, the message source is the satellite, the channel is outer space together with the hardware that sends and receives the data, and the receiver is the ground station on Earth. (Of course, messages travel from Earth to the satellite as well.) For the compact disc, the message is the voice, music, or data to be placed on the disc, the channel is the disc itself, and the receiver is the listener.
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- Information
- Fundamentals of Error-Correcting Codes , pp. 1 - 52Publisher: Cambridge University PressPrint publication year: 2003