On Packing Designs with Block Size $5$ and Indexes $3$ and $6$

Abstract

Let $V$ be a finite set of order $v$. A $(v,\kappa ,\lambda )$ packing design of index $\lambda$ and block size $u$ is a collection of $u$-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, $\sigma (v,\kappa ,\lambda )$, in a packing design. It is well known that $\sigma (v,\kappa ,\lambda )\le [\frac{v}{\kappa }[\frac{v-1}{\kappa -1}\lambda ]]=\psi (v,\kappa ,\lambda )$, where $[x]$ is the largest integer satisfying $x\ge [x]$. It is shown here that $\sigma (v,5,3)=\psi (v,5,3)$ for all positive integers $v\ge 5$ with the possible exceptions of $v=43$ and that $\sigma (v,5,3)=\psi (v,5,3)$ for all positive integers $v=1,5,9,17(\mathrm{mod}20)$ and $\sigma (v,5,3)=\psi (v,5,3)–1$ for all positive integers $v\equiv 13(\mathrm{mod}20)$ with the possible exception of $v=17,29,33,49$.