Doubly and orthogonally resolvable quadruple systems

Abstract

A Steiner system (X,β), denoted S_λ(t,k,v), is a set X of points, of cardinality v, and a collection β of k-subsets of X called blocks, with the property that every t-subset of X is contained in precisely λ blocks. A quadruple system of order v is a Steiner system S₁(3,4,v). A triple (X,β,γ) is called an (s,μ)-resolvable system if for some s<t, it is a partition of an S_λ(t,k,v) system (X,β) into subsystems (X,γ_i), each of which is an S_μ(s,k,v) system with γ=γ₁|γ₂|…|γ_c being a partition of β. An (s,μ)-resolvable system (X,β,γ) is (s,μ)(r,v)-doubly resolvable if each (X,γ_i) is (r,v)-resolvable. Two (1,1)-resolvable Steiner systems (X,β,γ) and (X,β,γ′) are orthogonal if |γ_i ∩ γ′_j|⩽1 for all i and j. This paper contains constructions of (2,3)(1,1)-doubly resolvable quadruple systems of orders v = 20,32,44, 68,80 and 104; a(2,3)-resolvable quadruple system of order 128; and some sets of mutually orthogonal resolutions of quadruple systems of all the above orders.

This work forms part of the author’s research towards a Ph.D. at the University of Newcastle.