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

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