Optimal Codes Correcting Localized Deletions

TL;DR

This paper introduces explicit codes capable of correcting localized deletions within a window, generalizing burst deletion correction, with asymptotically optimal redundancy for certain parameters.

Contribution

The authors develop novel explicit codes that efficiently correct localized deletions with near-optimal redundancy, extending beyond traditional burst deletion correction.

Findings

Codes can correct up to k localized deletions within a window.

Redundancy is logarithmic in message length, asymptotically optimal for certain k.

Codes are efficiently encodable and decodable.

Abstract

We consider the problem of constructing codes that can correct deletions that are localized within a certain part of the codeword that is unknown a priori. Namely, the model that we study is when at most $k$ deletions occur in a window of size $k$, where the positions of the deletions within this window are not necessarily consecutive. Localized deletions are thus a generalization of burst deletions that occur in consecutive positions. We present novel explicit codes that are efficiently encodable and decodable and can correct up to $k$ localized deletions. Furthermore, these codes have $\log n+\mathcal{O}(k \log^2 (k\log n))$ redundancy, where $n$ is the length of the information message, which is asymptotically optimal in $n$ for $k=o(\log n/(\log \log n)^2)$.