Optimal Codes Correcting Localized Deletions
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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+𝒪(k log^2 (klog n)) redundancy, where n is the length of the information message, which is asymptotically optimal in n for k=o(log n/(loglog n)^2).
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