On the spectrum of some cyclically and transitively oriented H-designs

This paper investigates oriented ℱ-designs on complete uniform hypergraphs of rank 3, focusing in particular on the spectrum of existence and on the construction of some cyclically and transitively oriented P(3)(2, 4)-designs and P(3)(1, 5)-designs, namely BCP(3)(2, 4)-designs and BCP(3)(1, 5)-designs for cyclically oriented ones, BTP(3)(2, 4)-designs and BTP(3)(1, 5)-designs for transitively oriented ones. In the appendix, we provide the Python code to obtain the explicit realization of the BCP(3)(2, 4)-designs on v vertices. Moreover, the structure of this algorithm, with suitable modifications, can be generalized to the other three structures as well.