Erasure/List Random Coding Error Exponents Are Not Universally Achievable

Abstract

Abstract-We study the problem of universal decoding for unknown discrete memoryless channels in the presence of erasure/list option at the decoder, in the random coding regime. Specifically, we harness a universal version of Forney's classical erasure/list decoder developed in earlier studies, which is based on the competitive minimax methodology, and guarantees universal achievability of a certain fraction of the optimum random coding error exponents. In this paper, we derive an exact singleletter expression for the maximum achievable fraction. Examples are given in which the maximal achievable fraction is strictly less than unity, which imply that, in general, there is no universal erasure/list decoder which achieves the same random coding error exponents as the optimal decoder for a known channel. This is in contrast to the situation in ordinary decoding (without the erasure/list option), where optimum exponents are universally achievable, as is well known. It is also demonstrated that previous lower bounds derived for the maximal achievable fraction are not tight in general. We then analyze a generalized random coding ensemble which incorporate a training sequence, in conjunction with a suboptimal practical decoder ("plug-in" decoder), which first estimates the channel using the known training sequence, and then decodes the remaining symbols of the codeword using the estimated channel. One of the implications of our results, is setting the stage for a reasonable criterion of optimal training. Finally, we compare the performance of the "plug-in" decoder and the universal decoder, in terms of the achievable error exponents, and show that the latter is noticeably better than the former.Index Terms-Universal decoding, error exponents, erasure/list decoding, maximum-likelihood decoding, random coding, generalized likelihood ratio test, training sequence, plug-in decoder, channel uncertainty, competitive minimax.

Exact Error and Erasure Exponents for the Asymmetric Broadcast Channel

“…In a breakthrough, Somekh-Baruch and Merhav [13] derived the exact random coding exponents for erasure decoding. Recently, Huleihel et al [14] showed that the random coding exponent for erasure decoding is not universally achievable and established a simple relation between the total and undetected error exponents. Weinberger and Merhav [15] analyzed a simplified decoder for erasure decoding.…”

“…where the notation γ(Q U|YQY ) is consistent due to the fact that the function γ(Q, R 2 ) (see (14)) only depends on the marginal distribution Q UY . Therefore, by using a similar argument as that for Ψ b1 above, we can remove the nonconvex constraint β(Q) ≥ 0 in Φ * a1 due to Φ * a2 .…”

“…distribution defined in (1) where Q UY = Q U|YQY . Note that the notation γ(Q UY , R 2 ) is consistent since the function γ(Q, R 2 ) (see (14)) only depends on the marginal distribution Q UY . 2) For a given Q UX|Y , N m1 (Q UX|YQY ), m 1 ∈ M 1 are i.i.d.…”

Simplified Erasure/List Decoding

“…Nonetheless, it can be easily verified that this restriction is only needed for the analysis of E * l (R, T ), and so that analysis of E * e (R, T ) is valid for this list mode T < 0. Moreover, it can be shown that E * l (R, T ) = E * e (R, T )+T must be satisfied in general (which was shown in [6] only for the BSC), see [11,Lemma 1].…”

“…Over the years, research was aimed towards various extensions. For example, in [8,Chapter 10] [9], [10], [11] universal versions of erasure/list decoders, i.e., decoders that perform well even under channel uncertainty. In [12], the size of the list was gauged by its ρ-th moment, where ρ = 1 corresponds to the average list size, as in [1], and an achievable pair of error exponent and list size exponents was found.…”