Deletion Codes in the High-Noise and High-Rate Regimes

Abstract

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following tradeoffs (for any $ \varepsilon > 0$ ): 1) codes that can correct a fraction $1- \varepsilon $ of deletions with rate $ \mathop {\mathrm {poly}}\nolimits ( \varepsilon )$ over an alphabet of size $ \mathop {\mathrm {poly}}\nolimits (1/ \varepsilon )$ ; 2) binary codes of rate $1-\tilde {O}(\sqrt { \varepsilon })$ that can correct a fraction $ \varepsilon $ of deletions; and 3) Binary codes that can be list-decoded from a fraction $(1/2- \varepsilon )$ of deletions with rate $ \mathop {\mathrm {poly}}\nolimits ( \varepsilon )$ . This paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet. The above-mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors.