# On $q$-covering designs

## TL;DR

This paper improves upper bounds on the minimum size of certain $q$-covering designs in projective spaces by constructing new designs, advancing the understanding of their minimal configurations.

## Contribution

The paper introduces new constructions that lead to tighter upper bounds on the size of specific $q$-covering designs, extending previous results.

## Key findings

- Improved upper bounds for $ ext{C}_q(2n, 3, 2)$, $n
geq 4$
- Enhanced bounds for $ ext{C}_q(3n + 8, 4, 2)$, $n
geq 0$
- Refined bounds for $ ext{C}_q(2n, 4, 3)$, $n
geq 4$

## Abstract

A $q$-covering design $\mathbb{C}_q(n, k, r)$, $k \ge r$, is a collection $\mathcal X$ of $(k-1)$-spaces of $\mathrm{PG}(n-1, q)$ such that every $(r-1)$-space of $\mathrm{PG}(n-1, q)$ is contained in at least one element of $\mathcal X$ . Let $\mathcal{C}_q(n, k, r)$ denote the minimum number of $(k-1)$-spaces in a $q$-covering design $\mathbb{C}_q(n, k, r)$. In this paper improved upper bounds on $\mathcal{C}_q(2n, 3, 2)$, $n \ge 4$, $\mathcal{C}_q(3n + 8, 4, 2)$, $n \ge 0$, and $\mathcal{C}_q(2n,4,3)$, $n \ge 4$, are presented. The results are achieved by constructing the related $q$-covering designs.