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：本文考虑在NRT-度量下（即基础偏序集为等长链的不交并时）$\mathbb{F}_q^{s\times r}$中具有填充半径$R$的码，并建立了存在完美码时参数$s, r$和$R$的必要条件。更具体地，对于$r,s\geq 2$且$R\geq 1$，我们证明若存在非平凡完美码，则$(r+1)(R+1)\leq rs$。此外，我们探索了与背包问题的关联，并建立了满足$r>R$的完美码与满足$r=R$的完美码之间的对应关系。