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.
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 = 2 t + 1 , an odd number.
The repetition codes can detect up to [ n / 2 ] 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 P e = ∑ k = n + 1 2 n ( n k ) ϵ k ( 1 − ϵ ) ( n − k ) , where ϵ is the error over the transmission channel.
The code word is ( n , 1 ) , where n = 2 t + 1 , an odd number.
Calculate the d H Hamming weight of the repetition code.if d H ≤ t , decode code word to be all 0'sif d H ≥ t + 1 , decode code word to be all 1'sIn 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.