  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 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 be
chosen arbitrarily large and $N$ is the number of quadrature points. This
convergence rate is known to be best possible up to some $\log N$ 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 $\integer_b$.
