Majority logic decoding

Majority logic decoding is a method to decode Repetition codes, based on assumption that the largest occurrence of a symbol, was transmitted message. It bases the decision on the frequency of occurrence of any symbol in the given received code vector.

Theory

If we have a binary alphabet made of $0,1$ respectively, then we use $(n,1)$ repetition code, we have the input bit mapped to the codeword as a string of $n$ duplicated input bits We generally choose $n=2t+1$ an odd number.

The repetition codes, can correct up to $[n/2]$ errors. Also, decoding errors occur, when the more than, these specified errors occur. so the probability of error for a repetition code is given using, $P_{e}=\sum _{t={\frac {n+1}{2}}}^{n}{\begin{bmatrix}n\\t\\\end{bmatrix}}P_{e}^{(t)}(1-P_{e})^{(n-t)}$

Algorithm

Assumptions

You have a $(n,1)$ code word, where $n=2t+1$ an odd number.

• Calculate the $d_{H}$ Hamming weight of the Repetition code.
• if $d_{H}\leq t$, decode code word to be all 0's
• if $d_{H}\geq t+1$, decode code word to be all 1's

Example

If you had a $(n,1)$ code, with R=[1 0 1 1 0], then you would decode it as,

• $n=5,t=2$, $d_{H}=3$, so R'=[1 1 1 1 1]
• Hence the transmitted message bit, was 1.

References

1. Rice University, http://cnx.rice.edu/content/m0071/latest/