Codes Correcting a Burst of Deletions or Insertions

TL;DR

This paper develops new codes that efficiently correct bursts of deletions or insertions, closing the gap between theoretical lower bounds and existing code constructions, and extends the model to more general burst scenarios.

Contribution

It provides near-optimal code constructions for burst-deletion correction with minimized redundancy and extends the model to non-consecutive and size-limited deletion bursts.

Findings

Redundancy of the new codes is at most log(n)+(b-1)log(log(n))+b-log(b)

Extended models include at most b consecutive deletions and at most b deletions (not necessarily consecutive)

Equivalent insertion models are also studied and linked to burst deletion correction.

Abstract

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.