All Admissible \(3-(\nu, 4, \lambda)\) Directed Designs Exist

In a \(t-(\nu,k,\lambda)\) directed design, the blocks are ordered \(k\)-tuples and every ordered \(t\)-tuple of distinct points occurs in exactly \(\lambda\) blocks (as a subsequence). We show that a simple \(3-(\nu,4,2)\) directed design exists for all \(v\). This completes the proof that the necessary condition \(\lambda v\equiv 0 \pmod 2\) for the existence of a \(3-(\nu,4,\lambda)\) directed design is sufficient.