Domination Mappings into the Hamming Ball: Existence, Constructions, and Algorithms

TL;DR

This paper studies injective mappings from binary vectors to Hamming balls with a domination property, providing necessary and sufficient conditions, explicit constructions, and algorithms for their existence.

Contribution

It introduces the concept of domination mappings into Hamming balls, establishes conditions for their existence, and develops algorithms to determine their feasibility.

Findings

Necessary conditions for existence are identified.

Explicit constructions are provided for specific parameters.

A polynomial-time algorithm is developed for certain domination graphs.

Abstract

The Hamming ball of radius $w$ in $\{0,1\}^n$ is the set ${\cal B}(n,w)$ of all binary words of length $n$ and Hamming weight at most $w$. We consider injective mappings $\varphi: \{0,1\}^m \to {\cal B}(n,w)$ with the following domination property: every position $j \in [n]$ is dominated by some position $i \in [m]$, in the sense that "switching off" position $i$ in $x \in \{0,1\}^m$ necessarily switches off position $j$ in its image $\varphi(x)$. This property may be described more precisely in terms of a bipartite \emph{domination graph} $G = ([m] \cup [n], E)$ with no isolated vertices, for all $(i,j) \in E$ and all $x \in \{0,1\}^m$, we require that $x_i = 0$ implies $y_j = 0$, where $y = \varphi(x)$. 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. In this paper, we begin with simple necessary conditions for the existence of an $(m,n,w)$-domination mapping $\varphi: \{0,1\}^m \to {\cal B}(n,w)$. 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 $m \le 3w$. One of our main results herein is a proof that the trivial necessary condition $|{\cal B}(n,w)| \ge 2^m$ for the existence of an injection 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.