All DBIBDs with block size four exist

A directed balanced incomplete block design with parameters (v,b,r,k,λ*), is a balanced incomplete block design with parameters (v,b,r,k,2λ*), in which the blocks are regarded as ordered k-tuples and in which each ordered pair of elements occurs in λ* blocks. By generalizing results of Hanani, we show that the necessary conditions for the existence of these designs, when k = 4, are sufficient.