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 av-set(of points)and B is a collection of cyclic triples on X such that every ordered pair of distinctpoints from X appears in exactly one cyclic triple of B.The cyclic triple(a,b,c)contains the orderedpairs(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 associatedsemisymmetric Latin square is frame self-orthogonal.It is known that an FSOMTS(1~n)exists for alln≡1(mod 3)except n=10 and for all n≥15,n≡0(mod 3)with possible exception that n=18.Inthis paper,it is shown that(i)an FSOMTS(2~n)exists if and only if n≡0,1(mod 3)and n>5 withpossible exceptions n ∈{9,27,33,39};(ii)an FSOMTS(3~n)exists if and only if n≥4,with possibleexceptions that n ∈{6,14,18,19}.