  Frolov's cubature formula on the unit hypercube has been considered important
since it attains an optimal rate of convergence for various function spaces.
Its integration nodes are given by shrinking a suitable full rank
$\mathbb{Z}$-lattice in $\mathbb{R}^d$ and taking all points inside the unit
cube. The main drawback of these nodes is that they are hard to find
computationally, especially in high dimensions.In such situations, quasi-Monte
Carlo (QMC) rules based on digital nets have proven to be successful. However,
there is still no construction known that leads to QMC rules which are optimal
in the same generality as Frolov's.
  In this paper we investigate a polynomial analog of Frolov's cubature
formula, which we expect to be important in this respect. This analog is
defined in a field of Laurent series with coefficients in a finite field. A
similar approach was previously studied in [M.~B.~Levin. Adelic constructions
of low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue
5, 2010.].
  We show that our construction is a $(t,m,d)$-net, which also implies bounds
on its star-discrepancy and the error of the corresponding cubature rule.
Moreover, we show that our cubature rule is a QMC rule, whereas Frolov's is
not, and provide an algorithm to determine its integration nodes explicitly.
  To this end we need to extend the notion of $(t,m,d)$-nets to fit the
situation that the points can have infinite digit expansion and develop a
duality theory. Additionally, we adapt the notion of admissible lattices to our
setting and prove its significance.
