# Majority Logic Decoding - Theory

Theory

In a binary alphabet made of, if a repetition code is used, then each input bit is mapped to the code word as a string of -replicated input bits. Generally, an odd number.

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