On Packing Designs with Block Size \(5\) and Indexes \(3\) and \(6\)

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 \(\sigma(v, \kappa, \lambda) \leq [\frac{v}{\kappa}[\frac{v-1}{\kappa-1}\lambda]] = \psi(v, \kappa, \lambda)\), where \([ x ]\) is the largest integer satisfying \(x \geq [ x ]\). It is shown here that \(\sigma(v, 5, 3) = \psi(v, 5, 3)\) for all positive integers \(v \geq 5\) with the possible exceptions of \(v = 43\) and that \(\sigma(v, 5, 3) = \psi(v, 5, 3)\) for all positive integers \(v = 1, 5, 9, 17 \pmod{20}\) and \(\sigma(v, 5, 3) = \psi(v, 5, 3) – 1\) for all positive integers \(v \equiv 13 \pmod{20}\) with the possible exception of \(v = 17, 29, 33, 49\).