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 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 (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.