  We study the efficient construction of good polynomial lattice rules, which
are special instances of quasi-Monte Carlo (QMC) methods. The integration rules
obtained are of particular interest for the approximation of multivariate
integrals in weighted Walsh spaces. In particular, we study a construction
algorithm which assembles the components of the generating vector, which is in
this case a vector of polynomials over a finite field, of the polynomial
lattice rule in a component-wise fashion. We show that the constructed QMC
rules achieve the almost optimal error convergence order in the function spaces
under consideration and prove that the obtained error bounds can, under certain
conditions on the involved weights, be made independent of the dimension. We
also demonstrate that our alternative component-by-component construction,
which is independent of the underlying smoothness of the function space, can be
implemented relatively easily in a fast manner. Numerical experiments confirm
our theoretical findings.
