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

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 $P_e=\sum _{k=\frac{n+1}{2}}^n(\genfrac{}{}{0ex}{}{n}{k}){ϵ}^k(1-ϵ{)}^{(n-k)}$, where $ϵ$ 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/