  We consider the efficient construction of polynomial lattice rules, which are
special cases of so-called quasi-Monte Carlo (QMC) rules. These are of
particular interest for the approximate computation of multivariate integrals
where the dimension $d$ may be in the hundreds or thousands. We study a
construction method that assembles the generating vector, which is in this case
a vector of polynomials over a finite field, of the polynomial lattice rule in
a digit-by-digit (or, equivalently, coefficient-by-coefficient) fashion. As we
will show, the integration error of the corresponding QMC rules achieves
excellent convergence order, and, under suitable conditions, we can vanquish
the curse of dimensionality by considering function spaces equipped with
coordinate weights. The construction algorithm is based on a quality measure
that is independent of the underlying smoothness of the function space and can
be implemented in a fast manner (without the use of fast Fourier
transformations). Furthermore, we illustrate our findings with extensive
numerical results.
