1. Introduction. The family of low-discrepancy sequences introduced by Sobol’in his 1967 paper [19] has had a tremendous impact on the field of quasi-Monte Carlo methods. Two important ways in which this impact has taken place are: 1) it gave rise to several other important contributions giving variations, extensions, and generalizations of this construction and the concepts used to study it; 2) despite all the generalizations that have been proposed, this construction continues to play an important role in the application of quasi-Monte Carlo methods in practice, with implementations offered in many software/calculation packages [1, 9, 10, 11, 23, 24].

The goal of this work is to study generalizations of Sobol’sequences that preserve two fundamental properties of the original construction, namely:(a) one-dimensional projections that are (0, 1)-sequences, and (b) an easyto-implement column-by-column construction for the generating matrices, based on linear recurrences determined by monic irreducible polynomials over Fb, where b is a prime power. We compare this generalization—which, as announced in the title, we call “irreducible Sobol’sequences in prime power bases”—with other closely related families of digital (t, s)-sequences. In particular, we describe in detail how our construction is included in the larger family of generalized Niederreiter sequences introduced by Tezuka [20]. The paper is organized as follows. In Section 2, we recall different frameworks to build the low-discrepancy sequences that are relevant to this work. Section 3 is devoted to connections between Niederreiter sequences in base 2 and Sobol’sequences, with our generalization of the latter introduced in Section 4, along with a description of its relation to generalized Niederreiter