Improved Lower Bounds on the Domination Number of Hypercubes and Binary Codes with Covering Radius One

Abstract

A dominating set on an $n $-dimensional hypercube is equivalent to a binary covering code of length $n $ and covering radius 1. It is still an open problem to determine the domination number $\gamma(Q_n)$ for $ n\geq10$ and $ n\ne2^{k},2^{k}-1 $ ($k\in\mathbb{N} $). When $n$ is a multiple of 6, the best known lower bound is $\gamma(Q_n)\geq \frac{2^n}{n}$, given by Van Wee (1988). In this article, we present a new method using congruence properties due to Laurent Habsieger (1997) and obtain an improved lower bound $\gamma(Q_n)\geq \frac{(n-2)2^n }{n^2-2n-2}$ when $n$ is a multiple of 6.