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A description of channel coding, in particular linear block codes.

Channel coding is a viable method to reduce information rate through the channel and increase reliability. This goal isachieved by adding redundancy to the information symbol vector resulting in a longer coded vector of symbols that aredistinguishable at the output of the channel. Another brief explanation of channel coding is offered in Channel Coding and the Repetition Code . We consider only two classes of codes, block codes and convolutional codes .

Block codes

The information sequence is divided into blocks of length k . Each block is mapped into channel inputs of length n . The mapping is independent from previous blocks, that is,there is no memory from one block to another.

k 2 and n 5

00 00000
01 10100
10 01111
11 11011
information sequencecodeword (channel input)

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A binary block code is completely defined by 2 k binary sequences of length n called codewords.

C c 1 c 2 c 2 k
c i 0 1 n
There are three key questions,
  • How can one find "good" codewords?
  • How can one systematically map information sequences into codewords?
  • How can one systematically find the corresponding information sequences from a codeword, i.e. , how can we decode?
These can be done if we concentrate on linear codes and utilize finite field algebra.

A block code is linear if c i C and c j C implies c i c j C where is an elementwise modulo 2 addition.

Hamming distance is a useful measure of codeword properties

d H c i c j # of places that they are different
Denote the codeword for information sequence e 1 1 0 0 0 0 0 by g 1 and e 2 0 1 0 0 0 0 by g 2 ,, and e k 0 0 0 0 0 1 by g k . Then any information sequence can be expressed as
u u 1 u k i 1 k u i e i
and the corresponding codeword could be
c i 1 k u i g i
Therefore
c u G
with c 0 1 n and u 0 1 k where G g 1 g 2 g k , a k x n matrix and all operations are modulo 2.

In with

00 00000
01 10100
10 01111
11 11011
g 1 0 1 1 1 1 and g 2 1 0 1 0 0 and G 0 1 1 1 1 1 0 1 0 0

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Additional information about coding efficiency and error are provided in Block Channel Coding .

Examples of good linear codes include Hamming codes, BCH codes, Reed-Solomon codes, and many more. The rate of these codes is defined as k n and these codes have different error correction and error detection properties.

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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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