Balanced House-systems and nestings

A G-design is called balanced if the degree of each vertex x is a con- stant. A G-design is called strongly balanced if for every i = 1, 2, ..., h, there exists a constant C i such that d A i (x) = C i for every vertex x, where A i s are the orbits of the automorphism group of G on its vertex-set and d A i (x) of a vertex is the number of blocks containing x as an element of A i . We say that a G-design is simply balanced if it is balanced, but not strongly balanced. In this paper we determine the spectrum for simply balanced and strongly balanced House-systems. Further, we determine the spectrum for House-systems of all admis- sible indices nesting C 4 -systems.