Saturating linear sets in \(\mathrm{PG}(2,q^4)\) (Q6565506)

An \(\mathbb{F}_q\)-linear set \(L_U\subseteq\mathrm{PG}(k-1,q^m)\) is said to be \(\rho\)-saturating if any point in \(\mathrm{PG}(k-1,q^m)\) belongs to at least one subspace of projective dimension \(\rho-1\) spanned by \(\rho\) points of \(L_U.\) The main problem is the determination of the smallest rank of a \(\rho\)-saturating \(\mathbb{F}_q\)-linear set in \(\mathrm{PG}(k-1,q^m),\) denoted by \(s_{q^m/q}(k, \rho).\)\N\NIn [\textit{D. Bartoli} et al., Finite Fields Appl. 95, Article ID 102390, 27 p. (2024; Zbl 1533.05049)], the following bounds \(4\leq s_{q^4/q}(3, 2)\leq 5\) were proven, and when \(q\) is large and even \(s_{q^4/q}(3, 2) = 5. \) In this paper, the last equation is proven for any prime \(q.\) The two results towards this goal are: a 1-saturating linear set of rank 4 cannot be contained in a line; and a 1-saturating linear set needs to be scattered, with respect to the lines or scattered with the property that all the lines have weight at most two except one of weight three. As a consequence of the main result, it's shown that all the linear sets of rank 5 in \(\mathrm{PG}(2, q^4),\) different from a line, are 1-saturating.