Codeword or noise? Exact random coding exponents for slotted asynchronism

We consider the problem of slotted asynchronous coded communication, where in each time frame (slot), the transmitter is either silent or transmits a codeword from a given (randomly selected) codebook. The task of the decoder is to decide whether transmission has taken place, and if so, to decode the message. We derive the optimum detection/decoding rule in the sense of the best trade-off among the probabilities of decoding error, false alarm, and misdetection. For this detection/decoding rule, we then derive single-letter characterizations of the exact exponential rates of these three probabilities for the average code in the ensemble. It is shown that previously suggested decoders care in general strictly sub-optimal.