Coding against deletions in oblivious and online models

We consider binary error correcting codes when errors are deletions. A basic challenge concerning deletion codes is determining p (adv) 0 , the zero-rate threshold of adversarial deletions, defined to be the supremumof allp for which there exists a code familywith rate bounded away from0 capable of correcting a fraction p of adversarial deletions. A recent construction of deletion-correcting codes [3] shows that p (adv) 0 ≥ √ 2 − 1, and the trivial upper bound, p(adv) 0 ≤ 1 2 , is the best known. Perhaps surprisingly, we do not know whether or not p (adv) 0 = 1/2. In this work, to gain further insight into deletion codes, we explore two related errormodels: 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 (obliv) 0 equals 1. For online deletions, where the channel decides whether to delete bit xi based only on knowledge of bits x1x2 . . . xi , define the deterministic zero-rate threshold for online deletions p (on,d ) 0 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 (adv) 0 = 1 2 if and only if p (on,d ) 0 = 1 2 . ∗Research supported in part by NSF grant CCF-1422045. †Email: [email protected] ‡Email: [email protected]