On Covering Designs with Block Size $5$ and Index $4$

Abstract

A $(v,k,\lambda )$ covering design of order $v$, block size $k$, and index $\lambda$ is a collection of $k$-element subsets, called blocks of a set $V$ such that every $2$-subset of $V$ occurs in at least $\lambda$ blocks. The covering problem is to determine the minimum number of blocks in a covering design. In this paper we solve the covering problem with $k=5$ and $\lambda =4$ and all positive integers $v$ with the possible exception of $v=17,18,19,22,24,27,28,78,98$.