Improved Bounds for Codes over Trees
Yanzhi Li, Wenjie Zhong, Tingting Chen, Xiande Zhang
TL;DR
The paper advances the theory of codes over labelled trees by deriving two general upper bounds that extend known results to all $d\le n-2$, establishing $A(n,d)=O((C n)^{n-d})$ with $C\in[1/2,1)$ dependent on $d$, and refining bounds further in regimes where $d$ is near $n$. It also strengthens lower bounds, showing $A(n,d)=\Omega((c n)^{n-d})$ for $d=\delta n$ and proving that for any fixed $k$, $A(n,d)=\Omega(n^k)$ when $d=n-\frac{(k+1)\ln n}{\ln\ln n}$. The authors provide four explicit constructions, including $(n,\frac{(n-3)(n-9)}{9},n-4)$ and $(n,\frac{n(n-2)^2}{32},n-13)$ tree-codes, plus two small cases $(8,28,5)$ and $(11,35,8)$, demonstrating substantial code sizes close to the distance bounds. These results sharpen the asymptotics for tree-codes, highlight the potential for large codes even when $d$ is close to $n$, and open questions about the precise growth for general $d$ and the exact Theta-regime around $A(n,d)=\Theta(n^2)$.
Abstract
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 $d\leq n-2$, and $A(n,d)=Ω((cn)^{n-d})$ for any $d$ linear with $n$, where constants $c\in(0,1)$ and $C\in [1/2,1)$ depending on $d$. Previously, only $A(n,d)=O(n^{n-d-1})$ for fixed $d$ and $A(n,d)=Ω(n^{n-2d})$ for $d\leq n/2$ were known, while the upper bound is improved for any $d$ and the lower bound is improved for $d\geq 2\sqrt{n}$. Further, for any fixed integer $k$, we prove the existence of codes of size $Ω(n^k)$ when $n-d=o(n)$, and give explicit constructions of codes which show $A(n,n-4)=Ω(n^2)$ and $A(n,n-13)=Ω(n^3)$.