Improved Bounds for Codes over Trees

Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with pairwise distance at least $d$, denoted by $A(n,d)$, where the distance between any two labelled trees is the minimum number of edit edge operations in order to transform one tree to another. By various tools from graph theory and algebra, we show that when $n$ is large, $A(n,d)=O((Cn{)}^{n-d})$ for any $d\le n-2$, and $A(n,d)=\mathrm{\Omega }((cn{)}^{n-d})$ for any $d$ linear with $n$, where constants $c\in (0,1)$ and $C\in [1\mathrm{/}2,1)$ depending on $d$. Previously, only $A(n,d)=O(n^{n-d-1})$ for fixed $d$ and $A(n,d)=\mathrm{\Omega }(n^{n-2d})$ for $d\le n\mathrm{/}2$ were known, while the upper bound is improved for any $d$ and the lower bound is improved for $d\ge 2\sqrt{n}$. Further, for any fixed integer $k$, we prove the existence of codes of size $\mathrm{\Omega }(n^k)$ when $n-d=o(n)$, and give explicit constructions of codes which show $A(n,n-4)=\mathrm{\Omega }(n^2)$ and $A(n,n-13)=\mathrm{\Omega }(n^3)$.