Packings with Block Size Five and Index One: \(v \equiv 2 \pmod {4}\)

Abstract

We use the results on \(5\)-GDDs to obtain optimal packings with block size five and index one. In particular, we prove that if \(v \equiv 2, 6, 10 \pmod{20}\), there exists an optimal packing with block size five on \(v\) points with at most \(32\) possible exceptions. Furthermore, if \(v \equiv 14, 18 \pmod{20}\), there exists an optimal packing with block size five on \(v\) points with a finite (large) number of possible exceptions.