Halton-type sequences in rational bases in the ring of rational integers and in the ring of polynomials over a finite field

Abstract

The aim of this paper is to generalize the well-known Halton sequences from integer bases to rational number bases and to translate this concept of Halton-type sequences in rational bases from the ring of integers to the ring of polynomials over a finite field. These two new classes of Halton-type sequences are low-discrepancy sequences. More exactly, the first class, based on the ring of integers, satisfies the discrepancy bounds that were recently obtained by Atanassov for the ordinary Halton sequence, and the second class, based on the ring of polynomials over a finite field, satisfies the discrepancy bounds that were recently introduced by Tezuka and by Faure & Lemieux for the generalized Niederreiter sequences.