discusses block error-correcting codes that make a decision on the basis of minimum distance. That is, given a code consisting of s equally likely codewords of length n, for each received sequence v, the receiver selects the codeword w for which the distance d(w, v) is a minimum. We would like to prove that this scheme is “ideal” in the sense that the receiver always selects the codeword for which the probability of w given v, p1w|v2, is a maximum. Because all codewords are assumed equally likely, the codeword that maximizes p1w|v2 is the same as the codeword that maximizes p1v|w2.

a. In order that w be received as v, there must be exactly d(w, v) errors in transmission, and these errors must occur in those bits where w and v disagree. Let b be the probability that a given bit is transmitted incorrectly and n be the length of a codeword. Write an expression for p1v|w2 as a function of

b, d(w, v), and n.

Hint: The number of bits in error is d(w, v) and the number of bits not in error is n - d(w, v).

b. Now compare p1v|w12 and p1v|w22 for two different codewords w1 and w2 by calculating p1v|w12/p1v|w22.

c. Assume that 0 6 b 6