On the Covering Numbers $C_2(v,k,t),t>3$

Abstract

A $t$-$(v,k,\lambda )$ covering is a set of blocks of size $k$ such that every $t$-subset of a set of $v$ points is contained in at least $\lambda$ blocks. The cardinality of the set of blocks is the size of the covering. The covering number $C_{\lambda }(v,k,t)$ is the minimum size of a $t$-$(v,k,\lambda )$ covering. In this article, we find upper bounds on the size of $t$-$(v,k,2)$ coverings for $t=3,4$, $k=5,6$, and $v\le 18$. Twelve of these bounds are the exact covering numbers.