Exact Random Coding Exponents for Erasure Decoding

Abstract

Abstract-Random coding of a channel with an erasure option is studied. By analyzing the large deviations behavior of the code ensemble, we obtain exact single-letter formulas for the error exponents in lieu of Forney's lower bounds. The analysis technique we use is based on an enhancement and specialization of tools for assessing the moments of certain distance enumerators, that were recently used for determining the exponential behavior of other communication systems. We specialize our results to the binary symmetric channel case with uniform random coding distribution and derive an explicit expression for the error exponent which, unlike Forney's bounds, does not involve optimization over two parameters. We also establish the fact that for the binary symmetric channel case with uniform random coding distribution, the difference between the exact error exponent corresponding to the probability of undetected decoding error and the error exponent corresponding to the erasure event is equal to the threshold parameter. Numerical calculations for the binary symmetric channel with uniform random coding distribution indicate that in this case Forney's bound coincides with the exact random coding exponent.

The likelihood decoder: Error exponents and mismatch

“…We make use of type class enumerators, which have been shown to permit exponentially tight analyses in numerous source and channel coding problems (e.g. see [22]- [24]). …”

“…We proceed by applying standard exponentially tight steps based on the fact that there are only polynomially many terms in the summation [22]- [24]:…”

“…We proceed by characterizing the behavior of the expectation in (24) for the joint types P XY with P X = Q n and P Y = P Y , using properties of type class enumerators. The arguments follow those of [24], [25], and are based on the fact that N y ( P XY ) has a Binomial distribution with exponentially many terms (M −1 . = e nR ) and an exponentially small success probability (P (X, y) ∈ T n ( P XY ) .…”

“…with probability approaching one super-exponentially fast, and hence the expectation in (24) has an error exponent of…”

Erasure/List Random Coding Error Exponents Are Not Universally Achievable

“…As was previously mentioned, even for a known channel, only lower bounds for the exponents were obtained by Forney [8]. More recently, inspired by a statistical-mechanical point of view on random code ensembles, Somekh-Baruch and Merhav [18] have found exact expressions for the exponents of the optimal erasure/list decoder, by assessing the moments of certain type class enumerators. In this paper, we tackle again the problem of erasure/list channel decoding using similar methods, and derive an exact expression for ξ * (R, T ) with respect to the exact erasure/list exponents of a known channels found in [18].…”