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 $r=R$. Using this correspondence we prove the non-existence of non-trivial perfect codes also for $s=R+2$.

翻译：本文考虑在NRT度量下（即当底层偏序集为等长链的不交并时）具有填充半径$R$的$\mathbb{F}_q^{s\times r}$中的码，并建立了完美码存在性对参数$s,r$和$R$的必要条件。具体而言，对于$r,s\geq 2$且$R\geq 1$，我们证明若存在非平凡完美码，则$(r+1)(R+1)\leq rs$。我们还探讨了与背包问题的联系，并建立了$r>R$情况下完美码与$r=R$情况下完美码之间的对应关系。利用该对应关系，我们进一步证明了当$s=R+2$时非平凡完美码的不存在性。