On the fast component-by-component algorithm for polynomial lattice rules

We present some simplification and observations concerning the fast component-by-component algorithm for polynomial lattice rules. This algorithm was introduced by Nuyens and Cools in

D. Nuyens and R. Cools. Fast component-by-component construction, a reprise for different kernels. In H. Niederreiter and D. Talay, editors, Monte Carlo and Quasi-Monte Carlo Methods 2004, pages 371–385. Springer-Verlag, 2006. Cited on pp. 82, 153, 178. See also Chapter 6 of Dirk’s PhD Thesis.

Let and be the adic expansion of (unique in the sense that infinitely many of the are different from ). We use the notation as in the book. Assume that is a prime number. For the sake of definiteness, consider the mean square worst-case error