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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

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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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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