Perfect Octagon Quadrangle Systems with an upper C4-system and a large spectrum

An octagon quadrangle is the graph consisting of an 8-cycle (xi, x2,x8) with two additional chords: the edges {x\,x4} and {x5,xg}. An octagon quadrangle system of order v and index A [OQS] is a pair (X, H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of AKv defined on X. An octagon quadrangle system £ = (X, H) of order v and index A is said to be upper C4 — perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a yU,-fold 4-cycle system of order v; it is said to be upper strongly perfect, if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a yU,-fold 4-cycle system of order v and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a g-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible.