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2019-06-12T04:15:47Z

Created page with "In error detection and correction, '''majority logic decoding''' is a method to decode repetition codes, based on the assumption that the largest number of occurrences..."

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In [[error detection and correction]], '''majority logic decoding''' is a method to decode [[repetition code]]s, based on the assumption that the largest number of occurrences of a symbol was the transmitted symbol.

==Theory==
In a binary alphabet made of <math>0,1</math>, if a <math>(n,1)</math> repetition code is used, then each input bit is mapped to the [[code word]] as a string of <math>n</math>-replicated input bits. Generally <math>n=2t + 1</math>, an odd number.

The repetition codes can detect up to <math>[n/2]</math> transmission errors. Decoding errors occur when the more than these transmission errors occur. Thus, assuming bit-transmission errors are independent, the probability of error for a repetition code is given by <math> P_e = \sum_{k=\frac{n+1}{2}}^{n}
{n \choose k}
\epsilon^{k} (1-\epsilon)^{(n-k)}</math>, where <math>\epsilon</math> is the error over the transmission channel.

==Algorithm==
===Assumptions===
The code word is <math>(n,1)</math>, where <math>n=2t+1</math>, an odd number.

* Calculate the <math>d_H</math> [[Hamming weight]] of the repetition code.
* if <math>d_H \le t </math>, decode code word to be all 0's
* if <math>d_H \ge t+1 </math>, decode code word to be all 1's

==Example==
In a <math>(n,1)</math> code, if R=[1 0 1 1 0], then
it would be decoded as,
* <math>n=5, t=2</math>, <math>d_H = 3 </math>, so R'=[1 1 1 1 1]
* Hence the transmitted message bit was 1.

==Source==

[http://wikipedia.org/ http://wikipedia.org/]

[[Category:Error-correcting codes]]
==See Also on BitcoinWiki==
* [[Xriba]]
* [[Bitlumens]]
* [[Õpet Foundation]]
* [[Travelertoken]]
* [[HELIX Orange]]

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Admin: Created page with "In error detection and correction, '''majority logic decoding''' is a method to decode repetition codes, based on the assumption that the largest number of occurrences..."