Three-fold Kirkman Packing Designs \( \text{KPD}_3(\{4,s^*\},v) \)

A three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) is a three-fold resolvable packing with maximum possible number of parallel classes, each containing one block of size 3 and all other blocks of size 4. This article investigates the spectra of three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) for \( s = 5 \) and \( 6 \), and we show that it contains all positive integers \( v \equiv s – 4 \pmod{4} \) with \( v \geq 17 \) if \( s = 5 \), and \( v \geq 26 \) if \( s = 6 \).