New Classes of Group Divisible Designs with Block Size $4$ and Group Type $g^u{m}^1$

Abstract

In this paper, we show that there exist all admissible 4-GDDs of type $g^6{m}^1$ for $g\equiv 0(\mathrm{mod}6)$. For 4-GDDs of type $g^u{m}^1$, where $g$ is a multiple of 12, the most values of $m$ are determined. Particularly, all spectra of 4-GDDs of type $g^u{m}^1$ are attained, where $g$ is a multiple of 24 or 36. Furthermore, we show that all 4-GDDs of type $g^u{m}^1$ exist for $g=10,20,28,84$ with some possible exceptions.