Explicit Two-Deletion Codes With Redundancy Matching the Existential Bound

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 $\Omega(2^n/n^4)$. 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 $\Omega(2^n/n^{3+o(1)})$ that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.