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Multiplexing Zero-Error and Rare-Error Communications Over a Noisy Channel

Two independent data streams are to be transmitted over a noisy discrete memoryless channel with noiseless (ideal) feedback. Errors are tolerated only in the second stream, provided that they occur with vanishing probability. The rate of the error-free stream cannot, of course, exceed the channel’s zero-error feedback capacity, and nor can the sum of the streams’ rates exceed the channel’s Shannon capacity. Using a suitable coding scheme, these necessary conditions are shown to characterize all the achievable rate pairs. Planning for the worst channel behavior—as is needed to achieve zero-error communication—and planning for the typical channel behavior—as is needed to communicate near the Shannon limit—are thus not incompatible. It is further shown that feedback may be beneficial for the multiplexing problem even on channels on which it does not increase the zero-error capacity.