Improvement on the Discrepancy of (t, e, s)-Sequences

Recently, a notion of (t, e, s)-sequences in base b was introduced, where e = (e₁, ⋯ , e_s) 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 (T_ε , 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 = b^t/s! φ^si=1 b^ei - 1/2_ei lob b.