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 , p 2 , … n = p, p^2, … for all integers p ≥ 2 p ≥ 2 . This paper provides algorithms for the construction of generating vectors which are finitely extensible for n = p , p 2 , … n = p, p^2, … for all integers p ≥ 2 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 component-by-component construction of Nuyens and Cools. Analogous results are presented for generating vectors with small weighted star discrepancy.