  We study the numerical approximation of integrals over $\mathbb{R}^s$ with
respect to the standard Gaussian measure for integrands which lie in certain
Hermite spaces of functions. The decay rate of the associated sequence is
specified by a single integer parameter which determines the smoothness classes
and the inner product can be expressed via $L_2$ norms of the derivatives of
the function.
  We map higher order digital nets from the unit cube to a suitable subcube of
$\mathbb{R}^s$ via a linear transformation and show that such rules achieve,
apart from powers of $\log N$, the optimal rate of convergence of the
integration error.
