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 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 \log ^{4} n) time. The coefficient ck 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. [ABSTRACT FROM PUBLISHER]