Embeddings Of Directed BIBDs With Block Size Four

A transitively ordered k-tuple (a1,a2, ???, ak) is defined to be the set {(ai,aj,) : 1 3? + 1, and ?(v -1), ?(w - 1) = 0(mod3). So, to complete the existence problem of a DB?(4, ?; v) with arbitrary A, we only need to deal with the case ? ? {1,3}. It is proved in this paper that if v > 4w + 1, the above necessary conditions of the existence of a DB?(4,?; v) and a DB?(4, 3; v) are also sufficient. And when 3w + 1 < v < 4?, we distinguish the values of v and w, which satisfy the necessary conditions, into 12 cases depending on v, w module 12, and show that a DBW(4,1; v) exists except for one case.Yin XiaohuiDirected by Prof. Yin Jianxing...