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

中文翻译：

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纠正局部删除的最佳代码

我们考虑构造可纠正在先验未知的码字的特定部分内定位的删除的代码的问题。即，我们研究的模型是在大小为$ k $的窗口中最多出现$ k $个删除时，该窗口内的删除位置不一定是连续的。因此，局部缺失是在连续位置发生的猝发缺失的概括。我们提出了新颖的显式代码，它们可以有效地进行编码和解码，并且可以纠正高达$ k $的本地化删除。此外，这些代码具有$ \ log n + \ mathcal {O}（k \ log ^ 2（k \ log n））$冗余，其中$ n $是信息消息的长度，在$ n $中渐近最优为$ k = o（\ log n /（\ log \ log n）^ 2）$。