Majority logic decoding

Theory

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

The repetition codes can detect up to [n/2] 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 P_e = \sum_{k=\frac{n+1}{2}}^{n} {n \choose k} \epsilon^{k} (1-\epsilon)^{(n-k)}, where \epsilon is the error over the transmission channel.

Algorithm

Assumption: the code word is (n,1), where n=2t+1, an odd number.

• Calculate the d_H Hamming weight of the repetition code.
• if d_H \le t , decode code word to be all 0's
• if d_H \ge t+1 , decode code word to be all 1's

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

Example

In a (n,1) code, if R=[1 0 1 1 0], then it would be decoded 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, https://web.archive.org/web/20051205194451/http://cnx.rice.edu/content/m0071/latest/