  Quadrature rules using higher order digital nets and sequences are known to
exploit the smoothness of a function for numerical integration and to achieve
an improved rate of convergence as compared to classical digital nets and
sequences for smooth functions. A construction principle of higher order
digital nets and sequences based on a digit interlacing function was introduced
in [J. Dick, SIAM J. Numer. Anal., 45 (2007) pp.~2141--2176], which interlaces
classical digital nets or sequences whose number of components is a multiple of
the dimension.
  In this paper, we study the use of polynomial lattice point sets for
interlaced components. We call quadrature rules using such point sets {\em
interlaced polynomial lattice rules}. We consider weighted Walsh spaces
containing smooth functions and derive two upper bounds on the worst-case error
for interlaced polynomial lattice rules, both of which can be employed as a
quality criterion for the construction of interlaced polynomial lattice rules.
We investigate the component-by-component construction and the Korobov
construction as a means of explicit constructions of good interlaced polynomial
lattice rules that achieve the optimal rate of the worst-case error. Through
this approach we are able to obtain a good dependence of the worst-case error
bounds on the dimension under certain conditions on the weights, while
significantly reducing the construction cost as compared to higher order
polynomial lattice rules.
