Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.

翻译：饱和集是有限域上射影空间中的组合对象，过去三十年间得到了深入研究。它们与汉明度量下的编码覆盖问题密切相关。本文考虑这类集合的线性版本（近年提出），该版本与秩度量下的覆盖问题相关。该领域的主要问题在于：饱和线性集的秩可以小到何种程度，以及如何构造小秩的饱和线性集。近期，Bonini、Borello和Byrne给出了给定射影空间中饱和线性集秩的下界，并在某些情形下证明了其紧性。本文构造了达到该下界的饱和线性集，并揭示了饱和性质与线性集散落性之间的联系。论文最后部分展示了该下界非紧性的若干参数情形。