  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.
