On Covering Designs with Block Size $5$ and Index $6$

Abstract

Let $V$ be a finite set of order $v$. A $(v,\kappa ,\lambda )$ covering design of index $\lambda$ and block size $\kappa$ is a collection of $\kappa$-element subsets, called blocks, such that every $2$-subset of $V$ occurs in at least $\lambda$ blocks. The covering problem is to determine the minimum number of blocks, $\alpha (v,\kappa ,\lambda )$, in a covering design. It is well known that
$\alpha (v,\kappa ,\lambda )\ge \lceil \frac{v}{\kappa }\lceil \frac{v-1}{\kappa -1}\lambda \rceil \rceil =\varphi (v,\kappa ,\lambda )$
where $\lceil x\rceil$ is the smallest integer satisfying $x\le \lceil x\rceil$. It is shown here that
$\alpha (v,5,6)=\varphi (v,5,6)$ for all positive integers $v\ge 5$, with the possible exception of $v=18$.