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We consider the efficient construction of polynomial lattice rules, which arespecial cases of so-called quasi-Monte Carlo (QMC) rules. These are ofparticular interest for the approximate computation of multivariate integralswhere the dimension $d$ may be in the hundreds or thousands. We study aconstruction method that assembles the generating vector, which is in this casea vector of polynomials over a finite field, of the polynomial lattice rule ina digit-by-digit (or, equivalently, coefficient-by-coefficient) fashion. As wewill show, the integration error of the corresponding QMC rules achievesexcellent convergence order, and, under suitable conditions, we can vanquishthe curse of dimensionality by considering function spaces equipped withcoordinate weights. The construction algorithm is based on a quality measurethat is independent of the underlying smoothness of the function space and canbe implemented in a fast manner (without the use of fast Fouriertransformations). Furthermore, we illustrate our findings with extensivenumerical results.

Component-by-component digit-by-digit construction of good polynomial lattice rules in weighted Walsh spaces