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 approximate to 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 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.