Pure tetrahedral quadruple systems with index two

An oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n with index λ , denoted byTQSλ ( n ) , is a pair ( X , ℬ ) , where X is an n ‐set and ℬ is a set of oriented tetrahedra (blocks) such that every cyclic triple on X is contained in exactly λ members of ℬ . ATQSλ ( n ) is pure if there do not exist two blocks with the same vertex set. When λ = 1 , the spectrum of a pure TQS ( n ) has been completely determined by Ji. In this paper, we show that there exists a pureTQS2 ( n ) if and only if n ≡ 1 , 2( mod   3 )and n ≥ 7 . A corollary is that a simpleQS4 ( n ) also exists if and only if n ≡ 1 , 2( mod   3 )and n ≥ 7 .