On the Zero-Error Capacity Threshold for Deletion Channels

TL;DR

This paper investigates the zero-error capacity of deletion channels under adversarial deletions, exploring the threshold of deletion probability for non-zero capacity and connecting it to longest common subsequence problems.

Contribution

It introduces new approaches to determine the deletion probability threshold for zero-error capacity, linking it to the longest common subsequence problem.

Findings

Identifies the deletion probability threshold for non-zero zero-error capacity.

Establishes a connection between deletion channel capacity and longest common subsequence.

Proposes multiple methods to analyze the zero-error capacity in adversarial deletion scenarios.

Abstract

We consider the zero-error capacity of deletion channels. Specifically, we consider the setting where we choose a codebook ${\cal C}$ consisting of strings of $n$ bits, and our model of the channel corresponds to an adversary who may delete up to $pn$ of these bits for a constant $p$. Our goal is to decode correctly without error regardless of the actions of the adversary. We consider what values of $p$ allow non-zero capacity in this setting. We suggest multiple approaches, one of which makes use of the natural connection between this problem and the problem of finding the expected length of the longest common subsequence of two random sequences.