Irreducible Sobol’ sequences in prime power bases
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).