Construction of scrambled polynomial lattice rules over $\mathbb{F}_2$ with small mean square weighted $\mathcal{L}_2$ discrepancy

The $\mathcal{L}_2$ discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Wo\'{z}niakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted $\mathcal{L}_2$ discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field $\mathbb{F}_2$ whose scrambled point sets have small mean square weighted $\mathcal{L}_2$ discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of $N^{-2+\delta}$ for all $\delta>0$, where $N$ denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol' sequences.