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 t 2n , 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.

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.

No abstract