On the exact \(t\)-value of Niederreiter and Sobol' sequences
The paper studies several well-known families of \((t,s)\)-sequences. For recent survey articles on \((t,s)\)-sequences [see \textit{H. Niederreiter}, Finite Fields Appl. 11, No. 3, 578--600 (2005; Zbl 1087.11051) and Nets, \((t,s)\)-sequences, and codes. Keller, Alexander (ed.) et al., Monte Carlo and quasi-Monte Carlo methods 2006. Selected papers based on the presentations at the 7th international conference `Monte Carlo and quasi-Monte Carlo methods in scientific computing', Ulm, Germany, August 14--18, 2006. Berlin: Springer, 83--100 (2008; Zbl 1196.11110)]. First the authors study the exact \(t\)-value of Niederreiter sequences introduced in \textit{H. Niederreiter} [ J. Number Theory 30, No. 1, 51--70 (1988; Zbl 0651.10034)]. Then they analyze the exact \(t\)-value of generalized Niederreiter sequences, see \textit{S. Tezuka} [ACM Trans. Model. Comput. Simul. 3, No. 2, 99--107 (1993; Zbl 0846.11045)]. For a range of dimensions of practical interest they show that Niederreiter-Xing sequences, see [\textit{C. Xing} and \textit{H. Niederreiter}, Acta Arith. 73, No. 1, 87--102 (1995; Zbl 0848.11038) and Finite Fields Appl. 2, No. 3, 241--273 (1996; Zbl 0893.11029)], are demonstrably better than Sobol' sequences, see \textit{I. M. Sobol'} [U.S.S.R. Comput. Math. Math. Phys. 7, No. 4, 86--112 (1967); translation from Zh. Vychisl. Mat. Mat. Fiz. 7, 784--802 (1967; Zbl 0185.41103)], in terms of the exact \(t\)-value. Previously, such a conclusion was not possible since only upper bounds on the exact \(t\)-value of these sequences and a general lower bound for all \((t,s)\)-sequences were available.
scientific article; zbMATH DE number 5377198