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Frolov's cubature formula on the unit hypercube has been considered importantsince 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 unitcube. The main drawback of these nodes is that they are hard to findcomputationally, especially in high dimensions.In such situations, quasi-MonteCarlo (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 optimalin the same generality as Frolov's.  In this paper we investigate a polynomial analog of Frolov's cubatureformula, which we expect to be important in this respect. This analog isdefined in a field of Laurent series with coefficients in a finite field. Asimilar approach was previously studied in [M.~B.~Levin. Adelic constructionsof low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue5, 2010.].  We show that our construction is a $(t,m,d)$-net, which also implies boundson 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 isnot, 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 thesituation that the points can have infinite digit expansion and develop aduality theory. Additionally, we adapt the notion of admissible lattices to oursetting and prove its significance.

Digital net properties of a polynomial analogue of Frolov's construction