Logarithmically larger deletion codes of all distances

Abstract

The deletion distance between two binary words $u,v\in \{0,1{\}}^n$ 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 $k\ge 1$ fixed and $n$ going to infinity, a $k$-deletion code $C\subseteq \{0,1{\}}^n$ of maximum size satisfies $\hat{I}{\text{c◯}}_k(2^n/n^{2k})\le |C|\le O_k(2^n/n^k)$. We make the first asymptotic improvement to these bounds by showing that there exist $k$-deletion codes with size at least $\hat{I}{\text{c◯}}_k(2^n\log n/n^{2k})$. 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.