Erasure and undetected error probabilities in the moderate deviations regime

The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths n is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels, the ensemble performance of a sequence of random codes is also analyzed to demonstrate the optimality of the above-mentioned codes. The tradeoff between the code rate, undetected and total errors as well as the threshold in a generalized likelihood ratio test is characterized asymptotically. In particular, the code rate tends to the capacity of the channel at a rate slower than n^-1/2 corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. The proof techniques involve applications of a modified (or “shifted”) version of the Gärtner-Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.