On Packing Designs with Block Size $5$ and index $7\le \lambda \le 21$

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
$\sigma (\nu ,\kappa \lambda )\le \left[\frac{\nu }{\kappa }\left[\frac{(\nu -1)}{(\kappa -1)}\lambda \right]\right]=\mathrm{\Psi }(\nu ,\kappa ,\lambda )$, where $[x]$ is the largest integer satisfying $x\ge [x]$.

It is shown here that $\sigma (\nu ,5,\lambda )=\mathrm{\Psi }(\nu ,5,\lambda )–e$ for all positive integers $\nu \ge 5$ and $7\le \lambda \le 21$, where $e=1\text{ if }\lambda (\nu -1)\equiv 0(\mathrm{mod}\kappa -1)\text{ and }\lambda \nu \frac{(\nu -1)}{(\kappa -1)}\equiv 1(\mathrm{mod}\kappa )$ and $e=0$ otherwise with the following possible exceptions of $(\nu ,\lambda )$ = (28,7), (32,7), (44,7), (32,9), (28,11), (39,11), (28,13), (28,15), (28,19), (39,21).