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

## TL;DR

This paper establishes new upper bounds on the minimal size of saturating sets in projective planes and spaces of even dimension, applicable to all q, and connects these bounds to linear covering codes.

## Contribution

It provides the first general upper bounds for saturating sets in projective planes of any order and extends these results to higher-dimensional projective spaces using inductive methods.

## Key findings

- Upper bound for saturating sets in projective planes: s(2,q) ≤ sqrt((q+1)(3ln q + ln ln q + ln(3/4))) + sqrt(q/(3ln q)) + 3.
- Bounds are valid for all q, not just large q.
- Results are expressed in terms of linear covering codes.

## Abstract

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.