The Existence of FGDRP(3, gu)'s

By an FGDRP$(3,g^u)$, we mean a uniform frame $(X,\cal G,\cal A)$ of block size 3, index 2 and type $g^u$, where the blocks of $\cal{A}$ can be arranged into a $gu/3\times gu$ array. This array has the properties: (1) the main diagonal consists of $u$ empty subarrays of sizes $g/3\times g$; (2) the blocks in each column form a partial parallel class partitioning $X \setminus G$ for some $G\in \cal G$, while the blocks in each row contain every element of $X \setminus G$ $3$ times and no element of $G$ for some $G\in \cal{G}$. The obvious necessary conditions for the existence of an FGDRP$(3,g^u)$ are $u\geq 5$ and $g\equiv 0$ (mod 3). In this paper, we show that these conditions are also sufficient with the possible exceptions of $(g,u)\in \{(6,15),(9,18),(9,28),(9,34),(30,15)\}$.