Reduced access structures with four minimal qualified subsets on six participants

In this paper, we discuss a point about applying known decomposition techniques in their most general form. Three versions of these methods, which are useful for obtaining upper bounds on the optimal information ratios of access structures, are known as: Stinson's $λ$-decomposition, $(λ, ω)$-decomposition and $λ$-weighted-decomposition, where the latter two are generalizations of the first one. We continue by considering the problem of determining the exact values of the optimal information ratios of the reduced access structures with exactly four minimal qualified subsets on six participants, which remained unsolved in Marti-Farre et al.'s paper [Des. Codes Cryptogr. 61 (2011), 167-186]. We improve the known upper bounds for all the access structures, except four cases, determining the exact values of the optimal information ratios. All three decomposition techniques are used while some cases are handled by taking full advantage of the generality of decompositions.