New lower bounds on the fraction of correctable errors under list decoding in combinatorial binary communication channels

The aim of the paper is to revive and develop results of an unpublished manuscript of A.G. D'yachkov. We consider a discrete memoryless channel (DMC) and prove a theorem on the exponential expurgation bound for list decoding with fixed list size $L$. This result is an extension of the classical exponential error probability bound for optimal codes over a DMC to the list decoding model over a DMC. As applications of this result, we consider a memoryless binary symmetric channel (BSC) and a memoryless binary asymmetric channel ($\mathrm{Z}$-channel). For the both channels, we derive a lower bound on the fraction of correctable errors for zero-rate transmission over the corresponding channels under list decoding with a fixed list size $L$ at the channel output. For the $\mathrm{Z}$-channel, we obtain this bound for an arbitrary input alphabet distribution $(1-w,w)$; we also find the optimum value of the obtained bound and prove that the fraction of errors correctable by an optimal code tends to $1$ as the list size $L$ tends to infinity.