The Existence of FrGBT D(4, gu)′ s

If the blocks of a GDD$${(X, \mathcal{G}, \mathcal{A})}$$ with block size 4, index 3 and type gu can be arranged into a (gu)/4 × (gu) array, such that: (1) the main diagonal consists of u empty subarrays of size g/4 × g; (2) the blocks in each column form a partition of X\G for some $${G \in \mathcal{G}}$$, while the blocks in each row contains every element of X\G 3 times and no element of G for some $${G \in \mathcal {G}}$$, then the design is called a frame generalized balanced tournament design and denoted by FrGBT D(4, gu). The necessary conditions for the existence of such a design are $${u \geq 6}$$ and $${g \equiv 0}$$(mod 4). In this paper, the sufficiency of these conditions is proved with some possible exceptions.