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