Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $q$-covering designs   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.

5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn