The Existence of FGDRP(3, gu)'s

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