  We define a Walsh space which contains all functions whose partial mixed
derivatives up to order $\delta \ge 1$ exist and have finite variation. In
particular, for a suitable choice of parameters, this implies that certain
Sobolev spaces are contained in these Walsh spaces. For this Walsh space we
then show that quasi-Monte Carlo rules based on digital
$(t,\alpha,s)$-sequences achieve the optimal rate of convergence of the
worst-case error for numerical integration. This rate of convergence is also
optimal for the subspace of smooth functions. Explicit constructions of digital
$(t,\alpha,s)$-sequences are given hence providing explicit quasi-Monte Carlo
rules which achieve the optimal rate of convergence of the integration error
for arbitrarily smooth functions.
