Coding against deletions in oblivious and online models

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 $p_0^{(on,d)}=\frac{1}{2}$.