Octagon Quadrangle Systems nesting 4-kite-designs having equi-indices

An octagon quadrangle is the graph consisting of an 8-cycle $(x_1,x_2,\dots,x_8)$(x1,x2,…,x8) with two additional chords: the edges $\{x_1,x_4\}${x1,x4} and $\{x_5,x_8\}${x5,x8}. An octagon quadrangle system of order $v$v and index $\lambda$λ (OQS) is a pair $(X,H)$(X,H), where $X$X is a finite set of $v$v vertices and $H$H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of $\lambda K_v$λKv defined on $X$X. An octagon quadrangle system $\Sigma=(X,H)$Σ=(X,H) of order $v$v and index $\lambda$λ is said to be upper $C_4$C4-perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a $\mu$μ-fold 4-cycle system of order $v$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 $\mu$μ-fold 4-cycle system of order $v$v and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a $\rho$ρ-fold 8-cycle system of order $v$v. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible.