ENEE 739C: Advanced Topics in Signal Processing: Coding Theory

This procedure may be actually called minimum distance decoding, the name “maximum likelihood” is justified in a short while. Intuitively is seems natural to assume that this decoding is the best we can hope for in terms of minimizing the error rate. Note if a vector x ∈ C (say a binary code) is sent over a binary symmetric channel W , the probability that a vector y is received on its output P (y|x) = W(yx) = p(1 − p)n−w where w = d(x,y). If c is the decoding result, the probability of a decoding error equals 1−P (c|y). Therefore, the mapping ψ that minimizes the probability of error for the codeword ci is given by ψ(y) = ci ⇐⇒ P (ci|y) > P (cj |y) ∀j 6= i.