Loose 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph Kₙ^ (3) of order n has a set V of cardinality n as its vertex set and the set of all 3-element subsets of V as its edge set. For n 2, let Zₙ denote the set of integers modulo n. For m > 3, let LCₘ^ (3) denote the 3-uniform hypergraph with vertex set Z₂₌ and edge set \\{2i, 2i+1, 2i+2\: i \0, 1, 2, , m-1\\}. Any hypergraph isomorphic to LCₘ^ (3) is a 3-uniform loose m -cycle. Given hypergraphs K and H, a decomposition of K into H is a partition \E₁, E₂, , Eb\ of the edge set of K such that, for each i \1, 2, , b\, the subhypergraph induced by Eᵢ is isomorphic to H. We show that there exists a decomposition of Kₙ^ (3) into LC₆^ (3) if and only if n 12 and n 0, 1, 2, 9, 10, 18, 20, 28 or 29 36.