Saturating linear sets of minimal rank

TL;DR

This paper investigates the minimal rank of saturating linear sets in projective spaces over finite fields, providing constructions that meet known bounds and exploring their relation to scattered linear sets.

Contribution

It offers new constructions of saturating linear sets that achieve the lower bound on rank and links their properties to scattered linear sets.

Findings

Constructed saturating linear sets meeting the lower bound

Established a connection between saturating property and scatteredness

Identified parameters where the lower bound is not tight

Abstract

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.