o the definition of mutual information H(X; Y) to obtain: H(X;Y) = H(X) H(Y|X) H(X) = H(X; Y) + H(Y|X) (13.39) = H(X; +0 max H(X;) Y) = C, which indeed proves the necessary condition R < C. However, this demonstration does not constitute the actual converse proof of the CCT, since the theorem concerns codes that are only asymptotically zero-error codes, as the block length n is increased indefinitely. The second, and formally complete way of demonstrating the converse proof of the CCT is trickier, as it requires one to establish and use two more properties. The whole point may sound, therefore, overly academic, and it may be skipped, taking for granted that the converse proof exists. But the more demanding or curious reader might want to go the extra mile. To this intent, the demonstration of this converse proof is found in Appendix L. We shall next look at the practical interpretation of the CCT in the case of noisy, symmetric binary channels. The corresponding channel capacity as given by Eq. (13.5), is C=1 - f(e) = 1 + E log E + (1 - E) log(1 - E), where E is the noise parameter or, equivalently, the bit error rate (BER). If we discard the cases of the ideal or noiseless channel (E = f(e) = 0) and the useless channel (E = 0.5, f(e) = 1), we have 0 < lf(e)l < 1 and, hence, the capacity C < 1, or strictly less than one bit per channel use. According to the CCT, the code rate for error-free transmission must satisfy R < C, here < C < 1. Thus block codes (n, n) of any length n, for which R = n/n = 1 are not eligible. Block codes (n, k) of any length n, for which R = k/n > C are not eligible either. The eligible block codes must satisfy R = k/n < C. To provide an example, assume for instance E = 0.01, corresponding to BER = 10-3 We obtain C = 0.9985, hence the CCT condition is R = k/n < 0.9985 Consider now the following possibilities: First, according to Chapter 11, the smallest linear block code, (7, 4), has a length n = 7, and a rate R = 4/7 = 0.57. We note that although this code is consistent with the condition