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 VarshamovTenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with ≈ 2n/n codewords of length n, i.e., at most logn bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than nΩ(1). For any fixed k, we construct a binary code with ck logn redundancy that can be decoded from k deletions in Ok(n log 4 n) time. The coefficient ck can be taken to be O(k2 logk), 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.