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 $\approx 2^n/n$ 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 $c_k \log n$ 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.

The claim that linear $k$-bit deletion codes must have rate at most about $1/(k+1)$ made in the appendix of the previous version is false, and is retracted in this version.

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 $\approx 2^n/n$ 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 $c_k \log n$ 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.

We also note that among linear codes capable of correcting $k$ deletions, the $(k+1)$-fold repetition code is essentially the best possible.