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(e) Now consider random coding for the channel, as in the proof of the channel coding theorem. Assume that 2nR codewords X n (1), X n (2),. . ., X n (2nR ) are chosen uniformly over the 2n possible binary sequences of length n. One of these codewords is chosen and sent over the channel. The receiver looks at the received sequence and tries to nd a codeword in the code that is jointly typical with the received sequence. As argued above, this corresponds to nding a codeword X n (i) such that Y n X n (i) A(n) (Z). For a xed codeword x n (i), what is the probability that the received sequence Y n is such that (x n (i), Y n ) is jointly typical (f) Now consider a particular received sequence y n = 000000 . . . 0, say. Assume that we choose a sequence X n at random, uniformly distributed among all the 2n possible binary n-sequences. What is the probability that the chosen sequence is jointly typical with this y n [Hint: This is the probability of all sequences x n such that y n x n A(n) (Z).] (g) Now consider a code with 29 = 512 codewords of length 12 chosen at random, uniformly distributed among all the 2n sequences of length n = 25. One of these codewords, say the one corresponding to i = 1, is chosen and sent over the channel. As calculated in part (e), the received sequence, with high probability, is jointly typical with the codeword that was sent. What is the probability that one or more of the other codewords (which were chosen at random, independent of the sent codeword) is jointly typical with the received sequence [Hint: You could use the union bound, but you could also calculate this probability exactly, using the result of part (f) and the independence of the codewords.] (h) Given that a particular codeword was sent, the probability of error (averaged over the probability distribution of the channel and over the random choice of other codewords) can be written as Pr(Error|x n (1) sent) =
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