Simple Finite-Length Achievability and Converse Bounds for the Deletion Channel and the Insertion Channel

TL;DR

This work develops finite-length upper bounds on code size for synchronization-error channels, specifically the binary deletion channel with deletion probability $\delta$ and the binary insertion channel with insertion probability $\iota$, at given block length and target error $\varepsilon$. By introducing layer-oriented reference output distributions, the authors adapt the symbol-wise converse bound to a computable form and derive two tightening bounds: the max-oriented CVB (MO-CVB) and the layer-oriented CVB (LO-CVB). They provide explicit formulas for LO-CVB for deletion and insertion channels, including the embedding-number quantities $E(m,w)$ and $E_1(w,m)$, and validate the bounds with extensive numerical results showing LO-CVB often improves upon the conventional BEC bound. Additionally, the paper delivers a simple greedy achievability bound and discusses the normal approximation for comparison, offering practical finite-length benchmarks for synchronization-error channels relevant to DNA storage and sequencing.

Abstract

We develop upper bounds on code size for independent and identically distributed deletion (insertion) channel for given code length and target frame error probability. The bounds are obtained as a variation of a general converse bound, which, though available for any channel, is inefficient and not easily computable without a good reference distribution over the output alphabet. We obtain a reference output distribution for a general finite-input finite-output channel and provide a simple formula for the converse bound on the capacity employing this distribution. We then evaluate the bound for the deletion channel with a finite block length and show that the resulting upper bound on the code side is tighter than that for a binary erasure channel, which is the only alternative converse bound for this finite-length setting. Also, we provide the similar results for the insertion channel.

Figures (4)

• Figure 1: The optimal code peformance, the GAVB and the LO-CVB for the case of $n=1$.
• Figure 2: The LO-CVB for the deletion channel $(D_m^{(\delta)})^n$ versus the BEC bound and the normal approximation. The x-coordinate is equal to the total number of input bits $N=mn$.
• Figure 3: GAVB for channel $I_{12}^{(\iota)}$.
• Figure 4: The LO-CVB for the insertion channel $(I_m^{(\iota)})^n$ versus the normal approximation and the lower capacity bound rahmati13bounds. The x-coordinate is equal to the total number of input bits $N=mn$.