In this entry I discuss the definition of (digital) higher order nets and sequences and some possible simplifications of the notation.

Digital higher order nets and sequences have been introduced in

- [1] J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., 46, 1519–1553, 2008. doi: 10.1137/060666639 For a Matlab program generating higher order digital nets, numerical examples and plots of higher order digital nets see here.

whereas higher order nets have been introduced in

- [2] J. Dick and J. Baldeaux, Equidistribution properties of generalized nets and sequences. In: Proceedings of the MCQMC’08 conference, Montreal, Canada, P. L’Ecuyer and A. Owen (eds.), pp. 305–323, 2009. doi: 10.1007/978-3-642-04107-5_19 An earlier version can be found here.

There are several parameters occurring in the definition of higher order nets, namely and for higher order sequences we have the parameters

(Digital) higher order nets and sequences are point sets and sequences such that

where is the number of quadrature points and is the smoothness of the integrand Continue reading