Pair Covering Designs with Block Size $5$ with Higher Index -The case of $v$ even

Abstract

A $(v,k,\lambda )$ covering design is a set of $b$ blocks of size $k$ such that each pair of points occurs in at least $\lambda$ blocks, and the covering number $C(v,k,\lambda )$ is the minimum value of $b$ in any $(v,k,\lambda )$ covering design. For $k=5$ and $v$ even, there are 24 open cases with $2\le \lambda \le 21$, each of which is the start of an open series for $\lambda ,\lambda +20,\lambda +40,\dots$. In this article, we solve 22 of these cases with $\lambda \le 21$, leaving open $(v,5,\lambda )=(44,5,13)$ and $(44,5,17)$ (and the series initiated for the former).