On Covering Designs with Block Size $5$ and Index $7$

Abstract

Let $V$ be a finite set of order $v$. A $(v,k,\lambda )$ covering design of index $\lambda$ and block size $k$ is a collection of $k$-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, $\alpha (v,k,\lambda )$, in a covering design. It is well known that $\alpha (v,k,\lambda )\ge \lceil \frac{v}{k}\lceil \frac{v-1}{k-1}\lambda \rceil \rceil =\varphi (v,k,\lambda )$, where $\lceil x\rceil$ is the smallest integer satisfying $x\le \lceil x\rceil$. It is shown here that $\alpha (v,5,7)=\varphi (v,5,7)$ for all positive integers $v\ge 5$ with the possible exception of $v=22,28,142,162$.