Optimal Error-Detecting Codes for General Asymmetric Channels via Sperner Theory

Several communication models that are of relevance in practice are asymmetric in the way they act on the transmitted "objects". Examples include channels in which the amplitudes of the transmitted pulses can only be decreased, channels in which the symbols can only be deleted, channels in which non-zero symbols can only be shifted to the right (e.g., timing channels), subspace channels in which the dimension of the transmitted vector space can only be reduced, unordered storage channels in which the cardinality of the stored (multi)set can only be reduced, etc. We introduce a formal definition of an asymmetric channel as a channel whose action induces a partial order on the set of all possible inputs, and show that this definition captures all the above examples. Such a general approach allows one to treat all these different models in a unified way, and to obtain a characterization of optimal error-detecting codes for many interesting asymmetric channels by using Sperner theory.