Upper bounds on the smallest size of a saturating set in projective planes and spaces of even dimension

In a projective plane $\Pi_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\mathcal{S}$ is saturating (or dense) if any point of $\Pi_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$. Modifying an approach of [31], we proved the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\Pi_{q}$: \begin{equation*} s(2,q)\leq \sqrt{(q+1)\left(3\ln q+\ln\ln q +\ln\frac{3}{4}\right)}+\sqrt{\frac{q}{3\ln q}}+3. \end{equation*} The bound holds for all q, not necessarily large. By using inductive constructions, upper bounds on the smallest size of a saturating set in the projective space $\mathrm{PG}(N,q)$ with even dimension $N$ are obtained. All the results are also stated in terms of linear covering codes.