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

TL;DR

This paper improves the lower bounds on the domination number of hypercubes, which are equivalent to covering codes with radius one, using number theory techniques for specific dimensions.

Contribution

It introduces a new method based on congruence properties to establish tighter lower bounds for hypercube domination numbers when dimensions are multiples of six.

Findings

New lower bound: rom or n multiple of 6.

Enhanced bounds surpass previous estimates for certain dimensions.

Method leverages number theory to improve combinatorial bounds.

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.