Logarithmically larger deletion codes of all distances

The deletion distance between two binary words u , v ∈ {0, 1} n is the smallest k such that u and v share a common subsequence of length n - k . A set C of binary words of length n is called a k -deletion code if every pair of distinct words in C has deletion distance greater than k . In 1965, Levenshtein initiated the study of deletion codes by showing that, for k ≥ 1 fixed and n going to infinity, a k -deletion code C ⊆ {0, 1} n of maximum size satisfies Ω k (2 n / n 2 k ) ≤ | C | ≤ O k (2 n / n k ). We make the first asymptotic improvement to these bounds by showing that there exist k -deletion codes with size at least Ω k (2 n log n / n 2 k ). Our proof is inspired by Jiang and Vardy's improvement to the classical Gilbert-Varshamov bounds. We also establish several related results on the number of longest common subsequences and shortest common supersequences of a pair of words with given length and deletion distance.