Perfect Tetra-Dodecahedron Systems

A tetra-dodecahedron is the hypergraph D = (X,E), uniform of rank 4, having 8 vertices and 4 edges, such that: X = {x1, x2, ..., x8}, E = {E1,E2,E3,E4}whereE1 ={x1,x5,x6,x7},E2 ={x2,x5,x6,x8},E3 = {x3, x5, x7, x8}, E4 = {x4, x6, x7, x8}. A tetra-dodecahedron system of order v and index ρ [TDS] is a pair Σ = (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint tetra-dodecagons (called blocks) which partitions the edge set of ρK(4), the complete hypergraph of order v, uniform of rank 4, defined in X. A tetra- dodecagon system of order v is said to be perfect [PTDS] if the col- lection of the inside K(3) contained in every block of Σ form a Steiner 4 quadruple systems [SQS]of order v and index μ. In this paper we de- termine completely the spectrum of PTDSs, with the non-restrictive condition that the inside SQS has index one.