Irreducible Sobol’ sequences in prime power bases

Henri Faure¹ ; Christiane Lemieux²

¹ Aix-Marseille Université CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille, France
² Department of Statistics and Actuarial Science University of Waterloo 200 University Avenue West Waterloo, ON N2L 3G1, Canada

Acta Arithmetica, Tome 173 (2016) no. 1, pp. 59-80

Résumé

Sobol’ sequences are a popular family of low-discrepancy sequences, in spite of requiring primitive polynomials instead of irreducible ones in later constructions by Niederreiter and Tezuka. We introduce a generalization of Sobol’ sequences that removes this shortcoming and that we believe has the potential of becoming useful for practical applications. Indeed, these sequences preserve two important properties of the original construction proposed by Sobol’: their generating matrices are non-singular upper triangular matrices, and they have an easy-to-implement column-by-column construction. We prove they form a subfamily of the wide family of generalized Niederreiter sequences, hence satisfying all known discrepancy bounds for this family. Further, their connections with Niederreiter sequences show these two families only have a small intersection (after reordering the rows of generating matrices of Niederreiter sequences in that intersection).

Détail

DOI : 10.4064/aa8226-1-2016

Keywords: sobol sequences popular family low discrepancy sequences spite requiring primitive polynomials instead irreducible later constructions niederreiter tezuka introduce generalization sobol sequences removes shortcoming believe has potential becoming useful practical applications indeed these sequences preserve important properties original construction proposed sobol their generating matrices non singular upper triangular matrices have easy to implement column by column construction prove form subfamily wide family generalized niederreiter sequences hence satisfying known discrepancy bounds family further their connections niederreiter sequences these families only have small intersection after reordering rows generating matrices niederreiter sequences intersection