# 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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what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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