  In this paper we investigate multivariate integration in weighted unanchored
Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we
employ higher order polynomial lattice point sets over $\mathbb{F}_{2}$ which
are randomly digitally shifted and then folded using the tent transformation.
We first prove the existence of good higher order polynomial lattice rules
which achieve the optimal rate of the mean square worst-case error, while
reducing the required degree of modulus by half as compared to higher order
polynomial lattice rules whose quadrature points are randomly digitally shifted
but not folded using the tent transformation. Thus we are able to restrict the
search space of generating vectors significantly. We then study the
component-by-component construction as an explicit means of obtaining good
higher 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 how
to obtain the fast component-by-component construction using the fast Fourier
transform. 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 polynomial
lattice rules are considered.
