On the Daróczy-Tsallis capacities of discrete channels

In the past there has been an extensive work on generalized entropies and generalized channel capacities. One of the first was Daroczy, who introduced new parameterized generalization of Shannon entropy, which reduces to the Shannon case if the parameter is set to one. A variant of this entropy, with a different normalization constant, was later proposed by Tsallis, who set it up as a basis for non-extensive statistical mechanics. Based on the generalized entropy, Daroczy introduced generalized mutual information which shares several important properties with the Shannon case, such as symmetry with respect to input/output channel distributions, non-negativity (if the parameter is greater than one) and obeying the chain rule. Daroczy also introduced a generalized channel capacity as the maximum of the generalized mutual information and derived expressions for the capacities of symmetric channel and binary symmetric channel as a special case.

In this paper we provide new expressions for Daroczy capacities of weakly symmetric channel, binary erasure channel and z-channel, extending the previous work by Daroczy. Similarly to the Shannon case, the capacity of weakly symmetric channel is expressed as the source entropy reduced by the entropy of the transition matrix row (scaled by appropriate constant), capacity of binary erasure channel is expressed as the q-average number of bits which can be recovered after transmission, while the capacity of z-channel is expressed in terms of q-logarithm and q-exponential of generalized binary entropy function. All the expressions are general and can be directly applied to Tsallis entropy, reducing to the Shannon capacity results in a limit case, when the parameter tends to one.