Skirting the $n$-tuples
Sam Adriaensen, Ferdinand Ihringer, William J. Martin, Ralihe R. Villagrán
TL;DR
The paper studies skirting sets in $\mathbb Z_q^n$, i.e., small subsets that skirt every $q$-ary $n$-tuple under the Hamming metric, equivalently posing the problem as finding the total domination number of the graph $G(n,q)$. It establishes a robust asymptotic framework: subadditivity of $f(n,q)$ leads to a limit $L_q$ and a growth constant $C_q=e^{L_q}$ with $f(n,q)=C_q^{(1+o(1))n}$, and provides initial bounds $1+\frac{1}{q-1} \le C_q \le q^{1/(q-1)}$ along with $C_q \ge \frac{q}{q-1}$. For small parameters, constructive bounds show $f(q,q)\le 2q-1$ via a concrete $S$, and ILP-assisted bounds give values for $f(n,3)$ and $f(n,4)$ up to $n\le8$. The introduction of skirting arrays $\mathsf{SA}(N;t,n,q)$ and their relation to covering arrays through $\mathsf{SAN}(t,n,q) \le \mathsf{CAN}(t,n,q) + 1 - (q-1)^t$ provides a new toolkit, illustrated by an explicit construction $\mathsf{SA}(19;4,20,4)$ that yields $f(20,20)\le 35$. The work lays groundwork for exact constant identification, improved bounds, and array-based constructions, with several open questions guiding future research.
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.