Csiszar and Narayan : Channel Capacity for a given Decoding

AbstructFor discrete memoryless channels { LV: X + y } , we consider decoders, possibly suboptimal, which minimize a metric defined additively by a given function d(2. y) 2 0. The largest rate achievable by codes with such a decoder is called the d-capacity C d ( W ) . The choice d ( 2 , y ) = 0 if and only if (iff) LV(yls) > 0 makes Cd(W) equal to the “zero undetected error” or “erasures-only” capacity C,, ( W). The graph-theoretic concepts of Shannon capacity 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 Csisz6r and Komer and Hui, is not tight in general, but Cd(W) > 0 iff this bound is positive. The “product space” improvement of the lower bound is considered, and a “product space characterization” of C,, (CI’) is obtained. We also determine the erasures-only (e.0.) 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.