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 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.