Error Exponents of Erasure/List Decoding Revisited via Moments of Distance Enumerators

TL;DR

This paper revisits error exponents for erasure/list decoding, introducing a method based on moments of distance enumerators that can tighten bounds and simplify analysis under certain conditions.

Contribution

It presents a new analysis technique using moments of distance enumerators that can match or improve upon Forney's error bounds with simpler optimization.

Findings

The new bound is at least as tight as Forney's bound.

Under symmetry conditions, the new bound involves only one parameter.

For the BSC, the optimal parameter can be found in closed form.

Abstract

The analysis of random coding error exponents pertaining to erasure/list decoding, due to Forney, is revisited. Instead of using Jensen's inequality as well as some other inequalities in the derivation, we demonstrate that an exponentially tight analysis can be carried out by assessing the relevant moments of a certain distance enumerator. The resulting bound has the following advantages: (i) it is at least as tight as Forney's bound, (ii) under certain symmetry conditions associated with the channel and the random coding distribution, it is simpler than Forney's bound in the sense that it involves an optimization over one parameter only (rather than two), and (iii) in certain special cases, like the binary symmetric channel (BSC), the optimum value of this parameter can be found in closed form, and so, there is no need to conduct a numerical search. We have not found yet, however, a numerical example where this new bound is strictly better than Forney's bound. This may provide an additional evidence to support Forney's conjecture that his bound is tight for the average code. We believe that the technique we suggest in this paper can be useful in simplifying, and hopefully also improving, exponential error bounds in other problem settings as well.