# 7.8 Channel coding

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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 $n=5$

$\mathrm{00}\to \mathrm{00000}$
$\mathrm{01}\to \mathrm{10100}$
$\mathrm{10}\to \mathrm{01111}$
$\mathrm{11}\to \mathrm{11011}$
information sequencecodeword (channel input)

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

$C=\{{c}_{1}, {c}_{2}, , {c}_{{2}^{k}}\}$
${c}_{i}\in \{0, 1\}^{n}$
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 $c_{i}\in C$ and $c_{j}\in C$ implies $c_{i}\mathop{\mathrm{xor}}c_{j}\in C$ where  is an elementwise modulo 2 addition.

Hamming distance is a useful measure of codeword properties

${d}_{H}(c_{i}, c_{j})=\text{# of places that they are different}$
Denote the codeword for information sequence ${e}_{1}=\left(\begin{array}{c}1\\ 0\\ 0\\ 0\\ \\ 0\\ 0\end{array}\right)$ by ${g}_{1}$ and ${e}_{2}=\left(\begin{array}{c}0\\ 1\\ 0\\ 0\\ \\ 0\\ 0\end{array}\right)$ by ${g}_{2}$ ,, and ${e}_{k}=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ \\ 0\\ 1\end{array}\right)$ by ${g}_{k}$ . Then any information sequence can be expressed as
$u=\left(\begin{array}{c}{u}_{1}\\ \\ {u}_{k}\end{array}\right)=\sum_{i=1}^{k} {u}_{i}{e}_{i}$
and the corresponding codeword could be
$c=\sum_{i=1}^{k} {u}_{i}{g}_{i}$
Therefore
$c=uG$
with $c=\{0, 1\}^{n}$ and $u\in \{0, 1\}^{k}$ where $G=\left(\begin{array}{c}{g}_{1}\\ {g}_{2}\\ \\ {g}_{k}\end{array}\right)$ , a $k$ x $n$ matrix and all operations are modulo 2.

In with

$\mathrm{00}\to \mathrm{00000}$
$\mathrm{01}\to \mathrm{10100}$
$\mathrm{10}\to \mathrm{01111}$
$\mathrm{11}\to \mathrm{11011}$
${g}_{1}=\left(\begin{array}{c}0\\ 1\\ 1\\ 1\\ 1\end{array}\right)$ and ${g}_{2}=\left(\begin{array}{c}1\\ 0\\ 1\\ 0\\ 0\end{array}\right)$ and $G=\begin{pmatrix}0 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 0\\ \end{pmatrix}$

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 $\frac{k}{n}$ and these codes have different error correction and error detection properties.

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