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 $\mathrm{P}\mathrm{G}(n-1,q)$ such that every $(r-1)$-space of $\mathrm{P}\mathrm{G}(n-1,q)$ 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 ${\mathcal{C}}_q(2n,4,3)$, $n \ge 4$,arepresented\mathrm{.}Theresultsareachievedbyconstructingtherelated$q$-covering designs.