Codeword or Noise? Exact Random Coding Exponents for Joint Detection and Decoding

We consider the problem of coded communication, where in each time frame, 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 tradeoff 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 probabilities for the average code in the ensemble. It is shown that previously proposed decoders are in general strictly suboptimal.