# Constructing a new class of low-discrepancy sequences by using the β-adic transformation

## Abstract

It is well known that low-discrepancy sequences and their discrepancy play essential roles in quasi Monte Carlo methods [5]. In this paper, a new class of low-discrepancy sequences Nβ is constructed by using the ergodic theoretical transformation which is called β-adic transformation [7,8]. Here, β is a real number greater than 1. When β is an integer greater than 2, Nβ becomes the classical van der Corput sequence in base β. Therefore, the class Nβ can be regarded as a generalization of the van der Corput sequence. It is shown that for some special β, the discrepancy of this sequence decreases in the fastest order O(N 1logN). We give the numerical results of discrepancy of Nβ for some βs. Pagès [6] also generalized van der Corput sequence in a different direction by using an ergodic transformation. © 1998 IMACS/Elsevier Science B.V.