Polynomial Lattice Point Sets

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Polynomial lattice point sets are special types of (t, m, s)-nets as introduced by H. Niederreiter in the 1980s. Quasi-Monte Carlo rules using them as underlying nodes are called polynomial lattice rules. In their overall structure polynomial lattice rules are very similar to usual lattice rules due to E. Hlawka and N. M. Korobov. The main difference is that here one uses polynomial arithmetic over a finite field instead of the usual integer arithmetic. In this overview paper we give a comprehensive review of the research on polynomial lattice rules during the last decade. We touch on topics like extensible polynomial lattice rules, higher order polynomial lattice rules and the weighted discrepancy of polynomial lattice point sets. Furthermore we compare polynomial lattice rules with lattice rules and show what results for polynomial lattice rules also have an analog for usual lattice rules and vice versa.