Tripacking of Pairs by Quintuples The Case $v\equiv 2(mod4)$

Abstract

Let $V$ be a finite set of order $\nu$. A $(\nu ,\kappa ,\lambda )$ packing 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 most $\lambda$ blocks. The packing problem is to determine the maximum number of blocks, $\sigma (\nu ,\kappa ,\lambda )$, in a packing design. It is well known that $\sigma (\nu ,\kappa ,\lambda )<[\frac{\nu }{\kappa }[\frac{(\nu -1)}{\kappa (\kappa -1)}]]=\psi (\nu ,\kappa ,\lambda )$, where $[x]$ is the largest integer satisfying $x\ge [x]$. It is shown here that if $v\equiv 2(\mathrm{mod}4)$ and $\nu \ge 6$ then $\sigma (\nu ,5,3)=\psi (\nu ,5,3)$ with the possible exception of $v=38$.