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 2n/n4+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(2n/n4). 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(2n/n3+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.