Improvements on the star discrepancy of (t,s)-sequences

The term low-discrepancy sequences is widely used to refer to s-dimensional sequences X satisfying the bound D∗(N,X) ≤ cs(log N)s + O((log N)s−1), where D∗ denotes the usual star discrepancy. In this article, we are concerned with (t, s)-sequences in base b, one of the most famous families of low-discrepancy sequences along with Halton sequences. The constants cs for (t, s)-sequences were first computed by Sobol’ and Faure in special cases and then achieved in general form by Niederreiter in the eighties. Then, quite recently, Kritzer improved these constants for s ≥ 2 by a factor 1 2 for an odd base and b 2(b+1) for an even base b ≥ 4. Our aim is to further improve the result in the case of an even base by a factor 2(b+1) b ( b−1 b )s−1 (s ≥ 2), hence obtaining the ratio 3 2s−1 for b = 2. Combining this estimate with best known t-values from the database MinT, we obtain new values for Niederreiter-Xing sequences where base 2 recovers supremacy on base 3 (in Kritzer’s paper). Our proof relies on a method of Atanassov to bound the discrepancy of Halton sequences. We also investigate (t, 1)-sequences for which the approach of Kritzer does not work.