Optimal Codes Correcting Localized Deletions

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
$k=o(\log n/(\log\log n)^{2})$
.