Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions

In this paper, we give explicit constructions of point sets in the s-dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case error for numerically integrating high-dimensional periodic functions. In the classical measure P-alpha of the worst-case error introduced by Korobov, the convergence, for every even integer alpha >= 1, is of O( N- (min(alpha,d))(logN)(s alpha-2)), where d is a parameter of the construction which can be chosen arbitrarily large and N is the number of quadrature points. This convergence rate is known to be the best possible up to some logN factors. We prove the result for the deterministic and also a randomized setting. The construction is based on a suitable extension of digital (t, m, s)-nets over the finite field Zb.