Efficient Low-Redundancy Codes for Correcting Multiple Deletions

We consider the problem of constructing binary codes to recover from <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>–bit deletions with efficient encoding/decoding, for a fixed <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. The single deletion case is well understood, with the Varshamov–Tenengolts–Levenshtein code from 1965 giving an asymptotically optimal construction with <inline-formula> <tex-math notation="LaTeX">$\approx ~2^{n}/n$ </tex-math></inline-formula> codewords of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, i.e., at most <inline-formula> <tex-math notation="LaTeX">$\log n$ </tex-math></inline-formula> bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than <inline-formula> <tex-math notation="LaTeX">$n^{\Omega (1)}$ </tex-math></inline-formula>. For any fixed <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, we construct a binary code with <inline-formula> <tex-math notation="LaTeX">$c_{k} \log n$ </tex-math></inline-formula> redundancy that can be decoded from <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> deletions in <inline-formula> <tex-math notation="LaTeX">$O_{k}(n \log ^{4} n)$ </tex-math></inline-formula> time. The coefficient <inline-formula> <tex-math notation="LaTeX">$c_{k}$ </tex-math></inline-formula> can be taken to be <inline-formula> <tex-math notation="LaTeX">$O(k^{2} \log k)$ </tex-math></inline-formula>, which is only quadratically worse than the optimal, non-constructive bound of <inline-formula> <tex-math notation="LaTeX">$O(k)$ </tex-math></inline-formula>. We also indicate how to modify this code to allow for a combination of up to <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> insertions and deletions. We also note that among <italic>linear</italic> codes capable of correcting <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> deletions, the <inline-formula> <tex-math notation="LaTeX">$(k+1)$ </tex-math></inline-formula>-fold repetition code is essentially the best possible.

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