The Existence of FrGBT D(4, gu)′ s

If the blocks of a GDD with block size 4, index 3 and type g u 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 , while the blocks in each row contains every element of X \ G 3 times and no element of G for some , then the design is called a frame generalized balanced tournament design and denoted by FrGBT D (4, g u ). The necessary conditions for the existence of such a design are and (mod 4). In this paper, the sufficiency of these conditions is proved with some possible exceptions.