 correspondence between this unique input and the unique output triplet is called a fan. Thus in the most general case, inputs are said to be distinguishable if their fans are nonconfusable or do not overlap. We shall now use the concept of distinguishable inputs to revisit the CCT. Figure 13.5 illustrates the correspondence between the typical set of input sequences (2nH(X) elements) and the typical set of output sequences (2nH(Y) elements). It is seen that a fan of 2nH(Y|X) most likely (and equiprobable) output sequences of the typical set of Yn corresponds to any input sequence Xi of the typical set of X". Define M as the number of nonconfusable fans, i.e., the fans that are attached to distinguishable input sequences. The number of output sequences corresponding to the M distinguishable inputs is, thus, M2nH(Y|X) In the ideal case, where all inputs are distinguishable (or all fans are nonconfusable), we have M2nH(Y|X) = 2nH(Y) If some of the fans overlap, we have 12nH(X)X) The most general case, thus, corresponds to the condition (13.32) 14 A.A. Bruen and M.A Forcinito, Cryptography, Information Theory and Error-Correction (New York: John Wiley & Sons, 2005), Ch. 12. 13.3 Shannon's channel coding theorem or, equivalently, (13.33) where C = max H(X; Y) is the binary-channel capacity (nC being the capacity of the binary channel's nth extension, i.e., using messages of n-bit length). If one only chooses distinguishable input sequences to transmit messages at the information or code rate R, one has M = 2nR possible messages, which gives R = (log M)/n. Replacing this definition into Eq. (13.33) yields the condition for the channel rate: R C C. (13.34) The fundamental property expressed in Eq. (13.34) is that the channel capacity represents the maximum rate at which input messages can be made uniquely distinguishable, corresponding to accurate transmission. We note here that the term "accurate" does not mean 100% error free. This is because the M2nH(Y)X) nonconfusable fans of the M distinguishable input messages ar