We introduce the concept of a rank saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of $s_{q^m/q}(k,\rho)$, which is the minimum $\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank $\rho$-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $s_{q^m/q}(k,\rho)$ and evaluate it for certain values of $k$ and $\rho$. We give constructions of rank $\rho$-saturating systems suggested from geometry.

翻译：本文引入了秩饱和系统的概念，并概述了其与给定覆盖半径的秩度量码之间的对应关系。我们考虑求解 $s_{q^m/q}(k,\rho)$ 值的问题，该值定义为 $\mathbb{F}_q^m$ 中 $k$ 维 $q$ 系统的最小 $\mathbb{F}_q$ 维数，且该系统是 $\rho$ 秩饱和的。这等价于秩度量中的覆盖问题。我们得到了 $s_{q^m/q}(k,\rho)$ 的上界和下界，并针对某些 $k$ 和 $\rho$ 值进行了计算。我们还给出了由几何学启发构造的 $\rho$ 秩饱和系统方案。