论文标题
改善了覆盖半径的高管和二元代码的统治数量的下限
Improved Lower Bounds on the Domination Number of Hypercubes and Binary Codes with Covering Radius One
论文作者
论文摘要
在$ n $维超过的$ n $二维上的主体设置等于二进制覆盖长度$ n $和覆盖半径1的二进制。确定$ n \ geq11 $和$ n \ ne2^{k}的支配数字$γ（q_n）$仍然是一个开放的问题。当$ n $是6个倍数时，最著名的下限是$γ（q_n）\ geq \ frac {2^n} {n} $，由van wee（1988）给出。在本文中，我们提出了一种新方法，使用laurent habsieger（1997）引起的一致性属性，并获得改进的下限$γ（q_n）\ geq \ frac {（n-2）2^n} {n^2-2n-2} $当$ n $是$ n $时，是倍数。
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 $γ(Q_n)$ 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 $γ(Q_n)\geq \frac{2^n}{n}$, 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 $γ(Q_n)\geq \frac{(n-2)2^n }{n^2-2n-2}$ when $n$ is a multiple of 6.
