Efficient Low-Redundancy Codes for Correcting Multiple Deletions
(1507.06175)Abstract
We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n{\Omega(1)}$. For any fixed $k$, we construct a binary code with $ck \log n$ redundancy that can be decoded from $k$ deletions in $Ok(n \log4 n)$ time. The coefficient $c_k$ can be taken to be $O(k2 \log k)$, which is only quadratically worse than the optimal, non-constructive bound of $O(k)$. We also indicate how to modify this code to allow for a combination of up to $k$ insertions and deletions.
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