Domination mappings into the hamming ball: Existence, constructions, and algorithms

Abstract

<p style='text-indent:20px;'>The Hamming ball of radius <inline-formula><tex-math id="M1">\begin{document}$ w $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \{0,1\}^n $\end{document}</tex-math></inline-formula> is the set <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{B}(n,w) $\end{document}</tex-math></inline-formula> of all binary words of length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> and Hamming weight at most <inline-formula><tex-math id="M5">\begin{document}$ w $\end{document}</tex-math></inline-formula>. We consider injective mappings <inline-formula><tex-math id="M6">\begin{document}$ \varphi : \{0,1\}^m \to \mathcal{B}(n,w) $\end{document}</tex-math></inline-formula> with the following <i>domination property:</i> every position <inline-formula><tex-math id="M7">\begin{document}$ j \in [n] $\end{document}</tex-math></inline-formula> is dominated by some position <inline-formula><tex-math id="M8">\begin{document}$ i \in [m] $\end{document}</tex-math></inline-formula>, in the sense that if position <inline-formula><tex-math id="M9">\begin{document}$ i $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M10">\begin{document}$ {\mathit{\boldsymbol{x}}} \in \{0,1\}^m $\end{document}</tex-math></inline-formula> is "switched off" (equal <i>zero</i>), then necessarily position <inline-formula><tex-math id="M11">\begin{document}$ j $\end{document}</tex-math></inline-formula> in its image <inline-formula><tex-math id="M12">\begin{document}$ \varphi({\mathit{\boldsymbol{x}}}) $\end{document}</tex-math></inline-formula> is switched off. This property may be described more precisely in terms of a bipartite <i>domination graph</i> <inline-formula><tex-math id="M13">\begin{document}$ G = \bigl([m] \cup [n], E\bigr) $\end{document}</tex-math></inline-formula> with no isolated vertices; for all <inline-formula><tex-math id="M14">\begin{document}$ (i,j) \in E $\end{document}</tex-math></inline-formula> and all <inline-formula><tex-math id="M15">\begin{document}$ {\mathit{\boldsymbol{x}}}\in \{0,1\}^m $\end{document}</tex-math></inline-formula>, we require that <inline-formula><tex-math id="M16">\begin{document}$ x_i = 0 $\end{document}</tex-math></inline-formula> implies <inline-formula><tex-math id="M17">\begin{document}$ y_j = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M18">\begin{document}$ {\mathit{\boldsymbol{y}}} = \varphi({\mathit{\boldsymbol{x}}}) $\end{document}</tex-math></inline-formula>. Although such domination mappings recently found applications in the context of coding for high-performance interconnects, to the best of our knowledge, they were not previously studied. The concept of domination mapping is thus interesting from both practical and combinatorial points of view.</p><p style='text-indent:20px;'>In this paper, we begin with simple necessary conditions for the existence of an <i><inline-formula><tex-math id="M19">\begin{document}$ (m,n,w) $\end{document}</tex-math></inline-formula>-domination mapping <inline-formula><tex-math id="M20">\begin{document}$ \varphi : \{0,1\}^m \to \mathcal{B}(n,w) $\end{document}</tex-math></inline-formula></i>. We then provide several explicit constructions of such mappings, which show that the necessary conditions are also sufficient when <inline-formula><tex-math id="M21">\begin{document}$ w = 1 $\end{document}</tex-math></inline-formula>, when <inline-formula><tex-math id="M22">\begin{document}$ w = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M23">\begin{document}$ m $\end{document}</tex-math></inline-formula> is odd, or when <inline-formula><tex-math id="M24">\begin{document}$ m \leqslant 3w $\end{document}</tex-math></inline-formula>. One of our main results herein is a proof that the trivial necessary condition <inline-formula><tex-math id="M25">\begin{document}$ | \mathcal{B}(n,w)| \geqslant 2^m $\end{document}</tex-math></inline-formula> is, in fact, sufficient for the existence of an <inline-formula><tex-math id="M26">\begin{document}$ (m,n,w) $\end{document}</tex-math></inline-formula>-domination mapping whenever <inline-formula><tex-math id="M27">\begin{document}$ m $\end{document}</tex-math></inline-formula> is sufficiently large. We also present a polynomial-time algorithm that, given any <inline-formula><tex-math id="M28">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M29">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M30">\begin{document}$ w $\end{document}</tex-math></inline-formula>, determines whether an <inline-formula><tex-math id="M31">\begin{document}$ (m,n,w) $\end{document}</tex-math></inline-formula>-domination mapping exists for a domination graph with an equitable degree distribution.</p>