Super-polynomial accuracy of multidimensional randomized nets using the median-of-means

We study approximate integration of a function \(f\) over \([0,1]^s\) based on taking the median of \(2r-1\) integral estimates derived from independently randomized \((t,m,s)\)-nets in base \(2\). The nets are randomized by Matousek's random linear scramble with a digital shift. If \(f\) is analytic over \([0,1]^s\), then the probability that any one randomized net's estimate has an error larger than \(2^{-cm^2/s}\) times a quantity depending on \(f\) is \(O(1/\sqrt{m})\) for any \(c