A construction of low-discrepancy sequences involving finite-row digital (t,s)-sequences

This paper introduces a construction principle for generating matrices of digital sequences over a finite field $\mathbb{F }_q$ , which is based on sequences of polynomials and their representations in terms of powers of nonconstant polynomials. For the most basic polynomial sequence, $(x^r)_{r\ge 0}$ , the representations in terms of powers of linear polynomials yield, within this construction principle, the Pascal matrices, which consist of binomial coefficients and were earlier introduced by Faure for finite prime fields and by Niederreiter for finite field extensions. Generally, for binomial type sequences of polynomials an interesting relation between the generating matrices is worked out, and further examples of generating matrices are given, which contain combinatorial magnitudes as, e.g., binomial coefficients, Stirling numbers of the first kind, Stirling numbers of the second kind, and Bell numbers. Moreover, within this construction principle, explicit constructions of finite-row generating matrices of digital $(t,s)$ -sequences are presented, which were so far only known for $t$ equal to $0$ .