Constructions of general Polynomial Lattice Rules based on the Weighted Star Discrepancy

In this paper we study construction algorithms for polynomial lattice rules over arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration. Niederreiter obtained an existence result for polynomial lattice rules over arbi-trary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algo-rithms for polynomial lattice rules over an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy. In this work we provide construction algorithms for polynomial lattice rules over arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algo-rithm. We show how the search space of the Korobov-type algorithm can be reduced without sacricing the convergence rate, hence this algorithm is particularly fast. Our ndings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy