Improvement on the discrepancy of (t, e, s)-sequences

Abstract

Recently, a  notion of $(t; e; s)$-sequences in base $b$ was introduced, where $e =(e_1 ,..., e_s)$ is a positive integer vector, and their discrepancy bounds were obtained
based on the signed splitting method. In this paper, we rst propose a general
framework of $ ({\bf{T}}_{\mathcal{E}},{\mathcal{E}}, s)$-sequences, and present that it includes $ (\bf{T}, s)$-sequences
and $ (t; e; s)$-sequences as special cases. Next, we show that a careful analysis leads
to improvement on the discrepancy bound of a $ (t, e, s)$-sequence in an even base $b$. It follows that the constant in the leading term of the star discrepancy bound
is given by

$$

c^∗_s =

\frac{b^{t}} {s!}   \Pi\limits^s_ i=1    \frac{ b^{e_{i}} − 1} {2e_i log b}.

$$