The construction of good extensible rank-1 lattices

It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n = p, p2, . . . for all integers p ≥ 2. This paper provides algorithms for the construction of generating vectors which are finitely extensible for n = p, p2, . . . for all integers p ≥ 2. The proofs which show that our algorithms yield good extensible rank-1 lattices are based on a sieve principle. Particularly fast algorithms are obtained by using the fast componentby-component construction of Nuyens and Cools. Analogous results are presented for generating vectors with small weighted star discrepancy.