In this paper we consider codes in $\mathbb{F}_q^{s\times r}$ with packing radius $R$ regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) and we establish necessary condition on the parameters $s,r$ and $R$ for the existence of perfect codes. More explicitly, for $r,s\geq 2$ and $R\geq 1$ we prove that if there is a non-trivial perfect code then $(r+1)(R+1)\leq rs$. We also explore a connection to the knapsack problem and establish a correspondence between perfect codes with $r>R$ and those with $r=R$.
翻译:在本文中,我们考虑在NRT度量(即,当底质是同一长度的链条脱节时)用包装半径$\mathb{F ⁇ q ⁇ s\ times r $(美元)的编码,我们根据存在完美代码的参数确定必要的条件。对于美元,更明确地说,对于2美元和1美元,我们证明,如果存在非三重完美代码,那么(r+1)(R+1)(R+1)\leq r$(美元)。我们还探讨与Knapack问题的联系,并在美元与R$和美元之间的完美代码之间建立联系。