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 <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula><italic>-burst-deletion/insertion-correcting code</italic> if it can correct a burst of deletions/insertions of any <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically <inline-formula> <tex-math notation="LaTeX">$\log (n)+b-1$ </tex-math></inline-formula>, the redundancy of the best code construction by Cheng <italic>et al.</italic> is <inline-formula> <tex-math notation="LaTeX">$b(\log (n/b+1))$ </tex-math></inline-formula>. In this paper, we close on this gap and provide codes with redundancy at most <inline-formula> <tex-math notation="LaTeX">$\log (n) + (b-1)\log (\log (n)) +b -\log (b)$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula>-burst-deletion-correcting codes and extend the burst deletion model to two more cases: 1) a deletion burst of at most <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> consecutive bits and 2) a deletion burst of size at most <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> (not necessarily consecutive). We extend our code construction for the first case and study the second case for <inline-formula> <tex-math notation="LaTeX">$b=3,4$ </tex-math></inline-formula>.

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