A recursive construction of t-wise uniform permutations

We present a recursive construction of a (2t + 1)-wise uniform set of permutations on 2n objects using a (2t+1)-(2n,n,·) combinatorial design, a t-wise uniform set of permutations on n objects and a (2t + 1)-wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t-wise uniform set of permutations on n objects whose size is at most t²ⁿ, the first non-trivial construction of an infinite family of t-wise uniform sets for t≥4. If a non-trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction.