Efficient Low-Redundancy Codes for Correcting Multiple Deletions

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 $\approx 2^n/n$codewordsoflength$n$,i\mathrm{.}e\mathrm{.},atmost$\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{\mathrm{\Omega }(1)}$. For any fixed $k$, we construct a binary code with $c_k\log n$ redundancy that can be decoded from $k$ deletions in $O_k(n{\log }^{4}n)$ time. The coefficient $c_k$ can be taken to be $O(k^2\log k)$, 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.