Super-simple, pan-orientable and pan-decomposable GDDs with block size 4

In this paper we study ( 4 , 2 μ ) -GDDs of type g n possessing both the pan-decomposable property introduced by Granville, Moisiadis, Rees, On complementary decompositions of the complete graph, Graphs and Combinatorics 5 (1989) 57–61 and the pan-orientable property introduced by Grüttmüller, Hartmann, Pan-orientable block designs, Australas. J. Combin. 40 (2008) 57–68. We show that the necessary condition for a ( 4 , 2 μ ) -GDD satisfying both of these properties, namely (1) n ≥ 4 , μ g ( n − 1 ) ≡ 0 (mod 3 ), and (2) g − 1 , n are not both even if μ is odd are sufficient. When λ = 2 , our designs are super-simple. o determine the spectrum of ( 4 , 2 ) -GDDs which are super-simple and possess some of the decomposable/orientable conditions, but are not pan-decomposable or pan-orientable. In particular, we show that the necessary conditions for a super-simple directable ( 4 , 2 ) -GDD of type g n are sufficient.