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 ≈ 2n/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 nOmega(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 log4 n) time. The coefficient ck can be taken to be O(k2 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.