Quasi-Monte Carlo via linear shift-register sequences

Linear recurrences modulo 2 with long periods have been widely used for constructing (pseudo)random number generators. We use them for quasi-Monte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period length and to estimate the integral by the average value of the integrand over all vectors of successive output values produced by the small generator. We examine randomizations of this scheme, discuss criteria for selecting the parameters, and provide examples. This approach can be viewed as a polynomial version of lattice rules.