Coding against deletions in oblivious and online models

TL;DR

This paper investigates deletion error-correcting codes in different models, showing that codes can correct nearly all deletions in oblivious models and establishing equivalences between thresholds in adversarial and online deletion scenarios.

Contribution

It introduces the zero-rate thresholds for oblivious and online deletion models, proving that codes can handle nearly all deletions in the oblivious case and relating thresholds between models.

Findings

Codes exist with positive rate for any fraction less than 1 in oblivious deletions.

The zero-rate threshold for oblivious deletions is exactly 1.

Thresholds for adversarial and online deletion models are equivalent at 1/2.

Abstract

We consider binary error correcting codes when errors are deletions. A basic challenge concerning deletion codes is determining $p_0^{(adv)}$, the zero-rate threshold of adversarial deletions, defined to be the supremum of all $p$ for which there exists a code family with rate bounded away from 0 capable of correcting a fraction $p$ of adversarial deletions. A recent construction of deletion-correcting codes [Bukh et al 17] shows that $p_0^{(adv)} \ge \sqrt{2}-1$, and the trivial upper bound, $p_0^{(adv)}\le\frac{1}{2}$, is the best known. Perhaps surprisingly, we do not know whether or not $p_0^{(adv)} = 1/2$. In this work, to gain further insight into deletion codes, we explore two related error models: oblivious deletions and online deletions, which are in between random and adversarial deletions in power. In the oblivious model, the channel can inflict an arbitrary pattern of $pn$ deletions, picked without knowledge of the codeword. We prove the existence of binary codes of positive rate that can correct any fraction $p < 1$ of oblivious deletions, establishing that the associated zero-rate threshold $p_0^{(obliv)}$ equals $1$. For online deletions, where the channel decides whether to delete bit $x_i$ based only on knowledge of bits $x_1x_2\dots x_i$, define the deterministic zero-rate threshold for online deletions $p_0^{(on,d)}$ to be the supremum of $p$ for which there exist deterministic codes against an online channel causing $pn$ deletions with low average probability of error. That is, the probability that a randomly chosen codeword is decoded incorrectly is small. We prove $p_0^{(adv)}=\frac{1}{2}$ if and only if $p_0^{(on,d)}=\frac{1}{2}$.