Covering problems in coding theory are closely related to finite geometry through the interpretation of the columns of parity-check matrices as point sets in finite vector spaces. Motivated by the recent notion of generalized covering radii of linear codes introduced by Elimelech, Firer and Schwartz, we develop a geometric framework for these parameters. We introduce $(ρ,t)$-saturating sets and show that they are precisely the finite-geometric counterparts of linear codes whose $t$-th generalized covering radius is at most $ρ$. We study the structure of these sets and show that the extremal case $ρ=t$ coincides with the notion of $t$-strong blocking sets. Thus, $(ρ,t)$-saturating sets interpolate between classical saturating sets and strong blocking sets. We provide several equivalent formulations, including affine and dual Grassmannian criteria, derive lower bounds on their size, and give constructions from strong blocking sets, graphs and projective configurations.

翻译：编码理论中的覆盖问题与有限几何密切相关，通过将校验矩阵的列解释为有限向量空间中的点集。受Elimelech、Firer和Schwartz最近引入的线性码广义覆盖半径概念的启发，我们为这些参数发展了一个几何框架。我们引入了$(ρ,t)$-饱和集，并证明它们恰好是$t$阶广义覆盖半径不超过$ρ$的线性码的有限几何对应对象。我们研究这些集的结构，并证明极端情况$ρ=t$与$t$-强阻塞集的概念一致。因此，$(ρ,t)$-饱和集在经典饱和集和强阻塞集之间起到插值作用。我们提供了若干等价表述，包括仿射和对偶格拉斯曼准则，推导了其大小的下界，并给出了来自强阻塞集、图和射影配置的构造。