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

The Hamming ball of radius $w$ in $\{0,1\}^n$ is the set $\mathcal{B}(n,w)$ of all binary words of length $n$ and Hamming weight at most $w$. We consider injective mappings $\varphi : \{0,1\}^m \to \mathcal{B}(n,w)$ with the following domination property: every position $j \in [n]$ is dominated by some position $i \in [m]$, in the sense that if position $i$ in ${\mathit{\boldsymbol{x}}} \in \{0,1\}^m$ is "switched off" (equal zero), then necessarily position $j$ in its image $\varphi({\mathit{\boldsymbol{x}}})$ is switched off. This property may be described more precisely in terms of a bipartite domination graph $G = \bigl([m] \cup [n], E\bigr)$ with no isolated vertices; for all $(i,j) \in E$ and all ${\mathit{\boldsymbol{x}}}\in \{0,1\}^m$, we require that $x_i = 0$ implies $y_j = 0$, where ${\mathit{\boldsymbol{y}}} = \varphi({\mathit{\boldsymbol{x}}})$. 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.

In this paper, we begin with simple necessary conditions for the existence of an $(m,n,w)$-domination mapping $\varphi : \{0,1\}^m \to \mathcal{B}(n,w)$. We then provide several explicit constructions of such mappings, which show that the necessary conditions are also sufficient when $w = 1$, when $w = 2$ and $m$ is odd, or when $m \leqslant 3w$. One of our main results herein is a proof that the trivial necessary condition $| \mathcal{B}(n,w)| \geqslant 2^m$ is, in fact, sufficient for the existence of an $(m,n,w)$-domination mapping whenever $m$ is sufficiently large. We also present a polynomial-time algorithm that, given any $m$, $n$, and $w$, determines whether an $(m,n,w)$-domination mapping exists for a domination graph with an equitable degree distribution.