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The $\mathcal{L}_2$ discrepancy is one of several well-known quantitativemeasures for the equidistribution properties of point sets in thehigh-dimensional unit cube. The concept of weights was introduced by Sloan andWo\'{z}niakowski to take into account the relative importance of thediscrepancy of lower dimensional projections. As known under the name ofquasi-Monte Carlo methods, point sets with small weighted $\mathcal{L}_2$discrepancy are useful in numerical integration. This study investigates thecomponent-by-component construction of polynomial lattice rules over the finitefield $\mathbb{F}_2$ 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 experimentsconfirm that the performance of our constructed polynomial lattice point setsis comparable or even superior to that of Sobol' sequences.

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