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

In this paper, we show that there exist all admissible 4-GDDs of type \(g^6m^1\) for \(g \equiv 0 \pmod{6}\). 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 \(g^um^1\) are attained, where \(g\) is a multiple of 24 or 36. Furthermore, we show that all 4-GDDs of type \(g^um^1\) exist for \(g = 10, 20, 28, 84\) with some possible exceptions.