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In this paper we investigate multivariate integration in weighted unanchoredSobolev spaces of smoothness of arbitrarily high order. As quadrature points weemploy higher order polynomial lattice point sets over $\mathbb{F}_{2}$ whichare randomly digitally shifted and then folded using the tent transformation.We first prove the existence of good higher order polynomial lattice ruleswhich achieve the optimal rate of the mean square worst-case error, whilereducing the required degree of modulus by half as compared to higher orderpolynomial lattice rules whose quadrature points are randomly digitally shiftedbut not folded using the tent transformation. Thus we are able to restrict thesearch space of generating vectors significantly. We then study thecomponent-by-component construction as an explicit means of obtaining goodhigher order polynomial lattice rules. In a way analogous to [J. Baldeaux, J.Dick, G. Leobacher, D. Nuyens, F. Pillichshammer, Numer. Algorithms, 59 (2012)403--431], we show how to calculate the quality criterion efficiently and howto obtain the fast component-by-component construction using the fast Fouriertransform. Our result generalizes the previous result shown by [L.L. Cristea,J. Dick, G. Leobacher, F. Pillichshammer, Numer. Math., 105 (2007) 413--455],in which the degree of smoothness is fixed at 2 and classical polynomiallattice rules are considered.

Constructing good higher order polynomial lattice rules with modulus of reduced degree