Frame self-orthogonal Mendelsohn triple systems of type hnhn

A Mendelsohn triple system (MTS) corresponds to an idempotent semisymmetric Latin square (quasigroup) of the same order. A holey MTS is called frame self-orthogonal, briefly FSOMTS, if its associated holey semisymmetric Latin square is frame self-orthogonal. In this paper, we use FSOMTS(hn)(hn) to denote an FSOMTS with n spanning holes of size h  . The existence of FSOMTS(hn)(hn) for h⩽3h⩽3 has been known with a few exceptions. We extend the existing results and determine the necessary and sufficient conditions for the existence of FSOMTS(hn)(hn) for any h and n with some possible exceptions.