Recently, a notion of (t, e, s)-sequences in base b was introduced, where e = (e1, ⋯ , es) is a positive integer vector, and their discrepancy bounds were obtained based on the signed splitting method. In this paper, we first propose a general framework of (Tε , ε, s)-sequences, and present that it includes (T, s)-sequences and (t, e, s)-sequences as special cases. Next, we show that a careful analysis leads to an asymptotic 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 = bt/s! φsi=1 bei - 1/2ei lob b.
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