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

In this paper we study (4,[email protected])-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 Gruttmuller, Hartmann, Pan-orientable block designs, Australas. J. Combin. 40 (2008) 57-68. We show that the necessary condition for a (4,[email protected])-GDD satisfying both of these properties, namely (1) n>=4, @mg(n-1)=0 (mod 3), and (2) g-1,n are not both even if @m is odd are sufficient. When @l=2, our designs are super-simple. We also 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.

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