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Informally, a \((t, w, v; m)\)-threshold scheme is a way of distributing partial information (chosen from a set of \(v\) shadows) to \(w\) participants, so that any \(t\) of them can easily calculate one of \(m\) possible keys, but no subset of fewer than \(t\) participants can determine the key. A perfect threshold scheme is one in which no subset of fewer than \(t\) participants can determine any partial information regarding the key. In this paper, we study the number \(M(t, w, v)\), which denotes the maximum value of \(m\) such that a perfect \((t, w, v; m)\)-threshold scheme exists. It has been shown previously that\(M(t, w, v) \leq (v-t+1)/(w-t+1)\), with equality occurring if and only if there is a Steiner system \(S(t, w, v)\) that can be partitioned into Steiner systems \(S(t-1, w, v)\). In this paper, we study the numbers \(M(t, w, v)\) in some cases where this upper bound cannot be attained. Specifically, we determine improved bounds on the values \(M(3, 3, v)\) and \(M(4, 4, v)\).