Channel detection in coded communication

We consider the problem of block-coded communication, where in each block, the channel law belongs to one
of two disjoint sets. The decoder is aimed to decode only
messages that have undergone a channel from one of the sets,
and thus has to detect the set which contains the underlying
channel. We begin with the simplified case where each of the
sets is a singleton. For any given code, we present the optimum
detection/decoding rule in the sense of the best trade-off among
the probabilities of decoding error, false alarm, and misdetection.
Then, we derive the exact single-letter characterization of the
random coding exponents for the optimal detector/decoder, as
well as an expurgated bound for low rates. We then extend the
random coding analysis to general sets of channels, and derive
the optimal detector/decoder under a worst case formulation of
the error probabilities, and derive its random coding exponents.