Codes Correcting a Burst of Deletions or Insertions

This paper studies codes that correct bursts of deletions. Namely, a code will be called a $b$-burst-deletion-correcting code if it can correct a deletion 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 also 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 $b=3,4$. The equivalent models for insertions are also studied and are shown to be equivalent to correcting the corresponding burst of deletions.