The spectrum for self-converse large set of pure Mendelsohn triple systems (Q2231728)

Let \(X\) be a finite set of \(v\) elements. A cyclic triple \(\langle x,y,z \rangle\) of distinct elements contains three ordered pairs \((x,y)\), \((y,z)\), and \((z,x)\). A Mendelsohn triple system MTS\((v,\lambda)\) is a pair \((X,\mathcal{A})\) where \(\mathcal{A}\) is a set of cyclic triples, called blocks, such that every ordered pair of \(X\) belongs to exactly \(\lambda\) blocks of \(\mathcal{A}\). An MTS\((v,\lambda)\) is called pure and denoted by PMTS\((v,\lambda)\) if \(\langle a,b,c \rangle \in \mathcal{A}\) implies \(\langle c,b,a \rangle \notin \mathcal{A}\). A large set of pure Mendelsohn triple systems, LPMTS\((v,\lambda)\) is a collection of \(\frac{v-2}{\lambda}\) disjoint pure Mendelsohn triple systems PMTS\((v,\lambda)\) on the same set of \(v\) elements. Every PMTS\((v,1)\) \((X,\mathcal{A}) \) has a converse \((X, \mathcal{A}^{-1})\) where \(\mathcal{A}^{-1} = \{ \langle z,y,x \rangle : \langle x,y,z \rangle \in\mathcal{A} \}\). The converse is also a PMTS\((v,1)\); a PMTS\((v,1)\) and its converse form a converse pair. An LPMTS\(^\ast(v)\) is a special LPMTS\((v,1)\) which contains exactly \(\frac{v-2}{2}\) converse pairs of PMTS\((v,1)\)s. These special large sets have application in the construction of maximum constant weight codes. In this paper, the authors investigate the existence of LPMTS\(^\ast(v)\) for \(v \equiv 6,10 \pmod{12}\). They use both direct and recursive constructions; their main recursive constructions use Mendelsohn candelabra systems. Their new existence results are then used to complete the spectrum and establish the following results. (1) There exists an LPMTS\(^\ast(v)\) if and only if \(v \equiv 0, 4 \pmod{6}\), \(v\geq 4\) and \(v\neq 6\). (2) There exists an LPMTS\((v,\lambda)\) with \(\lambda \equiv 2,4 \pmod{6}\) if and only if \(v\equiv 0,4 \pmod{6}\), \(v\geq 2\lambda + 2\), \(v\equiv 2 \pmod{\lambda}\).