Majority logic decoding

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 of a symbol was the transmitted symbol.

Theory

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In a binary alphabet made of , if a repetition code is used, then each input bit is mapped to the code word as a string of -replicated input bits. Generally , an odd number.

The repetition codes can detect up to transmission errors. Decoding errors occur when more than these transmission errors occur. Thus, assuming bit-transmission errors are independent, the probability of error for a repetition code is given by , where is the error over the transmission channel.

Algorithm

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Assumption: the code word is , where , an odd number.

  • Calculate the Hamming weight of the repetition code.
  • if , decode code word to be all 0's
  • if , decode code word to be all 1's

This algorithm is a boolean function in its own right, the majority function.

Example

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In a code, if R=[1 0 1 1 0], then it would be decoded as,

  • , , so R'=[1 1 1 1 1]
  • Hence the transmitted message bit was 1.

References

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  1. Rice University, https://web.archive.org/web/20051205194451/http://cnx.rice.edu/content/m0071/latest/