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In this paper we give explicit constructions of point sets in the $s$dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve theoptimal rate of convergence of the worst-case error for numerically integratinghigh dimensional periodic functions. In the classical measure $P_{\alpha}$ ofthe worst-case error introduced by Korobov the convergence is of$\landau(N^{-\min(\alpha,d)} (\log N)^{s\alpha-2})$ for every even integer$\alpha \ge 1$, where $d$ is a parameter of the construction which can bechosen arbitrarily large and $N$ is the number of quadrature points. Thisconvergence rate is known to be best possible up to some $\log N$ factors. Weprove the result for the deterministic and also a randomized setting. Theconstruction is based on a suitable extension of digital $(t,m,s)$-nets overthe finite field $\integer_b$.

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