A recursive construction of t-wise uniform permutations

TL;DR

This paper introduces a recursive method to construct t-wise uniform permutations on 2n objects, leveraging combinatorial designs and existing permutation sets, resulting in the first non-trivial infinite family for t ≥ 4.

Contribution

It presents a novel recursive construction method for t-wise uniform permutations using combinatorial designs, enabling infinite families for t ≥ 4.

Findings

Constructs t-wise uniform permutations with size at most t^2n

First non-trivial infinite family of t-wise uniform permutations for t ≥ 4

Potential for improved constructions if suitable combinatorial designs are found

Abstract

We present a recursive construction of a (2t + 1)-wise uniform set of permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) 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 t^2n, the first non-trivial construction of an infinite family of t-wise uniform sets for t \geq 4. If a non-trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction.