Some Constructions of Semiframes

A \((k, \lambda)\)-semiframe of type \(g^u\) is a group divisible design of type \(g^u\) \((\chi, \mathcal{G}, \mathcal{B})\), in which \(\mathcal{B}\) is written as a disjoint union \(\mathcal{B} = \mathcal{P} \cup \mathcal{Q} \) where \(\mathcal{P} \) is partitioned into partial parallel classes of \(\chi\) (with respect to some \(G \in \mathcal{G}\)) and \(\mathcal{Q} \) is partitioned into parallel classes of \(\chi\). In this paper, new constructions for these designs are provided with some series of designs with \(k = 3\). Cyclic semiframes are discussed. Finally, an application of semiframes is also mentioned.