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

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 <italic>worst case</italic> 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 <inline-formula> <tex-math notation="LaTeX">$ \varepsilon > 0$ </tex-math></inline-formula>): 1) codes that can correct a fraction <inline-formula> <tex-math notation="LaTeX">$1- \varepsilon $ </tex-math></inline-formula> of deletions with rate <inline-formula> <tex-math notation="LaTeX">$ \mathop {\mathrm {poly}}\nolimits ( \varepsilon )$ </tex-math></inline-formula> over an alphabet of size <inline-formula> <tex-math notation="LaTeX">$ \mathop {\mathrm {poly}}\nolimits (1/ \varepsilon )$ </tex-math></inline-formula>; 2) binary codes of rate <inline-formula> <tex-math notation="LaTeX">$1-\tilde {O}(\sqrt { \varepsilon })$ </tex-math></inline-formula> that can correct a fraction <inline-formula> <tex-math notation="LaTeX">$ \varepsilon $ </tex-math></inline-formula> of deletions; and 3) Binary codes that can be <italic>list-decoded</italic> from a fraction <inline-formula> <tex-math notation="LaTeX">$(1/2- \varepsilon )$ </tex-math></inline-formula> of deletions with rate <inline-formula> <tex-math notation="LaTeX">$ \mathop {\mathrm {poly}}\nolimits ( \varepsilon )$ </tex-math></inline-formula>. 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.

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