Skirting the $n$-tuples

TL;DR

This paper investigates the minimal size of a set in a Hamming space that dominates all points at maximum distance, providing bounds and constructions for the total domination number as the dimension grows.

Contribution

It introduces new bounds and constructions for the total domination number in Hamming spaces, advancing understanding of dominating sets in high-dimensional combinatorial structures.

Findings

Established that $f(n,q)$ grows exponentially with $n$

Provided bounds for the constants $C_q$ in the growth rate

Developed constructions for dominating sets in Hamming spaces

Abstract

Let $n\ge 2$ and $q\ge 2$ be given. The set $X = \mathbb Z_q^n$ is a metric space of diameter $n$ under the Hamming metric $d(\cdot,\cdot)$. We seek a smallest set $S\subseteq X$ that ``skirts'' every $q$-ary $n$-tuple in the sense that every $x\in X$ is at distance $n$ from at least one element of $S$. Thus we aim to compute the total domination number $f(n,q)$ of the graph $G(n,q)$ with vertex set $X$ and edge set $\{ xy \, \| \, d(x,y)=n\}$. We provide constructions and bounds for this number, establishing $f(n,q) = C_q^{(1+o(1))n}$ for some constants $2=C_2>C_3 \geq \cdots$ which we are only able to estimate at the present time.