Menu
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
\(\sigma(\nu, \kappa\lambda) \leq \left[\frac{\nu}{\kappa}\left[ \frac{(\nu-1)}{(\kappa-1)}\lambda\right]\right] = \Psi(\nu, \kappa, \lambda)\), where \([x]\) is the largest integer satisfying \(x \geq [x]\).
It is shown here that \(\sigma(\nu, 5, \lambda) = \Psi(\nu, 5, \lambda) – e\) for all positive integers \(\nu \geq 5\) and \(7 \leq \lambda \leq 21\), where \(e = 1\text{ if } \lambda(\nu-1) \equiv 0 \pmod{\kappa-1} \text{ and } \lambda\nu\frac{(\nu-1)}{(\kappa-1)} \equiv 1 \pmod{\kappa}\) 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).