Note on Existence and Non-Existence of Large Subsets of Binary Vectors with Similar Distances

TL;DR

This paper investigates the existence of large subsets of binary vectors with bounded distance ratios, establishing conditions under which such subsets cannot or can be found, depending on their size and weight.

Contribution

It provides new theoretical results characterizing when large subsets with similar distances exist or do not exist within sets of binary vectors.

Findings

No large subset with bounded distance ratio exists for certain parameters.

Existence of large subsets with bounded distance ratio is guaranteed under specific conditions.

Quantitative bounds on subset sizes relative to the original set are established.

Abstract

We consider vectors from $\{0,1\}^n$. The weight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\rm dr}(L):=\max \{\rho(x,y):\ x,y \in L\}/ \min \{\rho(x,y):\ x,y \in L,\ x\neq y\},$ where $\rho(x,y)$ is the Hamming distance between $x$ and $y$. We prove that (a) for every constant $\lambda>1$ there are no positive constants $\alpha$ and $C$ such that every set $K$ of at least $\lambda^p$ vectors with weight $p$ contains a subset $K'$ with $|K'|\ge |K|^{\alpha}$ and ${\rm dr}(K')\le C$, % even when $|K|\ge \lambda$, (b) For a set $K$ of vectors with weight $p$, and a constant $C>2$, there exists $K'\subseteq K$ such that ${\rm dr}(K')\le C$ and $|K'| \ge |K|^\alpha$, where $\alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil$.