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 formula_1, if a formula_2 repetition code is used, then each input bit is mapped to the code word as a string of formula_3-replicated input bits. Generally formula_4, an odd number. The repetition codes can detect up to formula_5 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 formula_6, where formula_7 is the error over the transmission channel. Algorithm. Assumption: the code word is formula_2, where formula_9, an odd number. This algorithm is a boolean function in its own right, the majority function. Example. In a formula_2 code, if R=[1 0 1 1 0], then it would be decoded as,