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Probability of j errors occurring an an n-digit codeword

Example 2: If we have an error correcting code which can correct 3 errors within a block length n of 10, what is the probability that the code cannot correct a received block if the per digit error probability is P e = 0.01?

Solution: The code cannot correct the received block if there are more than 3 errors. Thus:

P>3 errors = 1 - P(0 errors) - P(1 error) - P(2 errors) - P(3 errors).

[link] shows the component parts of this calculation.

Calculation of zero, 1, 2 and 3 errors in a 10-digit codeword with a per-digit P e of 0.01

Thus the probability that the code cannot correct a received block is then:

1 - 0.9043821 - 0.0913517 - 0.0041523 - 0.0001118 = 0.0000021.

This illustrates that the very low overall error remaining after correction of three errors is much less than the original probability of error in a single bit, P e = 0.01. Note also the need for high precision arithmetic (it may be an eight digit calculator is not good enough to calculate the answer to more than 1 significant figure).Note also in [link] the much lower probability of t + 1 errors occuring, compared to t errors, as is implied in FECC.

Group codes

Group codes are a special kind of block codes. They comprise a set of codewords, C1 … CN, which contain the all zeros codeword (e.g. 00000) and exhibit a special property called closure. This property means that if any two valid codewords are subject to a bit wise EX – OR operation then they will produce another valid codeword in the set.

The closure property means that to find the minimum Hamming distance, see below, all that is required is to compare all the remaining codewords in the set with the all zeros codeword instead of comparing all the possible pairs of codewords.

The saving gets bigger the longer the codeword. For example a code set with 100 codewords will require 100 comparisons for a Group code design, compared with 100+99+98+…+2+1, for a non-group code!

In Group codes the D min calculation is further simplified into calculating the minimum codeword weight or minimum number of 1 digits in a codeword in the set.

Nearest neighbour decoding

Nearest neighbour decoding assumes that the codeword nearest in Hamming distance to the received word is what was transmitted, as shown in Example 1 above. This inherently contains the assumption that the probability of a small number of t errors is greater than the probability of the larger number of t+1 errors, i.e that P e is small.

A nearest neighbour decoding table for a (n, k) = (5, 2) i.e. a 5-digit group code is shown in [link] . Recall that for an n = 5 bit codeword there are 2 5 = 32 unique patterns generated by all the possible combinations of the 5 digits.

Nearest neighbour decoding table for 5 bit code with 4 codewords implying 2 information bits

[link] starts by forming a table with the 4 codewords across the top row. All the single error patterns, which each only differ by one bit from each of the transmitted codewords, can be readily and uniquely assigned back to an error free codeword. Thus the next 5 rows represent these single errors in position 1 through 5 in each of the 4 codewords. Now we have a table up to this point with a total of 4 x 6 = 24 unique entries. Therefore this code is capable of correcting all these single errors.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Communications source and channel coding with examples. OpenStax CNX. May 07, 2009 Download for free at http://cnx.org/content/col10601/1.3
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