On the minimum number of completely 3-scrambling permutations

A family P={π₁,…,π_q} of permutations of n]={1,…,n} is completely k-scrambling Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x₁,…,x_k∈n], permutations π_i's in P produce all k! possible orders on π_i(x₁),…,π_i(x_k). Let N^*(n,k) be the minimum size of such a family. This paper focuses on the case k=3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison.