Frame self-orthogonal Mendelsohn triple systems

A Mendelsohn triple system of order v, MTS(v) for short, is a pair (X, B) where X is a v-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered pairs (a, b), (b, c) and (c, a). An MTS(v) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order v. An MTS(v) is called frame self-orthogonal, FSOMTS for short, if its associated semisymmetric Latin square is frame self-orthogonal. It is known that an FSOMTS(1(n)) exists for all n = 1 (mod 3) except n = 10 and for all n greater than or equal to 15, n = 0 (mod 3) with possible exception that n = 18. In this paper, it is shown that (i) an FSOMTS(2(n)) exists if and only if n = 0, 1 (mod 3) and n > 5 with possible exceptions n is an element of {9, 27, 33, 39}; (ii) an FSOMTS(3(n)) exists if and only if n greater than or equal to 4, with possible exceptions that n is an element of {6, 14, 18, 19}.