  In this paper we study quasi-Monte Carlo integration of smooth functions
using digital nets. We fold digital nets over $\mathbb{Z}_{b}$ by means of the
$b$-adic tent transformation, which has recently been introduced by the
authors, and employ such \emph{folded digital nets} as quadrature points. We
first analyze the worst-case error of quasi-Monte Carlo rules using folded
digital nets in reproducing kernel Hilbert spaces. Here we need to permit
digital nets with "infinite digit expansions," which are beyond the scope of
the classical definition of digital nets. We overcome this issue by considering
the infinite product of cyclic groups and the characters on it. We then give an
explicit means of constructing good folded digital nets as follows: we use
higher order polynomial lattice point sets for digital nets and show that the
component-by-component construction can find good \emph{folded higher order
polynomial lattice rules} that achieve the optimal convergence rate of the
worst-case error in certain Sobolev spaces of smoothness of arbitrarily high
order.
