Some Constructions of Semiframes

Abstract

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.