I-binomial scrambling of digital nets and sequences

The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owen's random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions. This paper considers a reduction of randomness required for Owen's random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owen's key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i-binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owen's scrambling also hold with the left i-binomial scrambling.