Codes Correcting a Burst of Deletions or Insertions

This paper studies codes that correct a burst of deletions or insertions. Namely, a code will be called a b-burst-deletion/insertion-correcting code if it can correct a burst of deletions/insertions of any b consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically log(n) + b - 1, the redundancy of the best code construction by Cheng et al. is b(log(n/b + 1)). In this paper, we close on this gap and provide codes with redundancy at most log(n) + (b - 1) log(log(n)) + b - log(b). We first show that the models of insertions and deletions are equivalent and thus it is enough to study codes correcting a burst of deletions. We then derive a non-asymptotic upper bound on the size of b-burst-deletion-correcting codes and extend the burst deletion model to two more cases: 1) a deletion burst of at most b consecutive bits and 2) a deletion burst of size at most b (not necessarily consecutive). We extend our code construction for the first case and study the second case for b = 3, 4.