论文标题
饱和系统和等级覆盖半径
Saturating systems and the rank covering radius
论文作者
论文摘要
我们介绍了等级饱和系统的概念，并将其对应关系概述为具有给定覆盖半径的等级代码。我们考虑找到$ s_ {q^m/q}（k，ρ）$的价值的问题，这是$ \ \ \ m athbb {f} _ {q^m}^k $ c $ p $ρ$ - ρ$ - ρ$ - c的最低$ \ mathbb {f} _q $ -dimension。这等同于等级度量中的覆盖问题。我们在$ s_ {q^m/q}（k，ρ）$上获得上限和下限，并以$ k $和$ρ$的某些值进行评估。我们提供从几何形状建议的等级$ρ$饱和系统的结构。
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 $s_{q^m/q}(k,ρ)$, which is the minimum $\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank $ρ$-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $s_{q^m/q}(k,ρ)$ and evaluate it for certain values of $k$ and $ρ$. We give constructions of rank $ρ$-saturating systems suggested from geometry.
