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

Abstract

A three-fold Kirkman packing design ${\text{KPD}}_3(\{4,s^{\ast }\},v)$ 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 ${\text{KPD}}_3(\{4,s^{\ast }\},v)$ for $s=5$ and $6$, and we show that it contains all positive integers $v\equiv s–4(\mathrm{mod}4)$ with $v\ge 17$ if $s=5$, and $v\ge 26$ if $s=6$.