Channel capacity for a given decoding metric

For discrete memoryless channels {W: X→Y} we consider decoders, possibly suboptimal, which minimize a metric defined additively by a given function d(x, y)⩾0. The largest rate achievable by codes with such a decoder is called the d-capacity C_d (W). The choice d(x, y)=0 if and only if (iff) W(y|x)>0 makes C _d(W) equal to the “zero undetected error” or “erasures-only” capacity C_eo(W). The graph-theoretic concepts of Shannon capacity (1956, 1974) and Sperner capacity are also special cases of d-capacity, viz. for a noiseless channel with a suitable {0, 1}-valued function d. We show that the lower bound on d-capacity given previously by Csiszar and Korner (1980), and Hui (1983), is not tight in general, but C_d(W)>0 iff this bound is positive. The “product space” improvement of the lower bound is considered,and a “product space characterization” of C_eo(W) is obtained. We also determine the erasures-only (e.o.) capacity of a deterministic arbitrarily varying channel defined by a bipartite graph, and show that it equals capacity. We conclude with a list of challenging open problems