Explicit two-deletion codes with redundancy matching the existential bound
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We give an explicit construction of length-n binary codes capable of correcting the deletion of two bits that have size 2^n/n^4+o(1). This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size Ω(2^n/n^4). Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size Ω(2^n/n^3+o(1)) that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.
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