The Discrepancy of Digital Nets

Digital sequences and nets are among the most popular kinds of low discrepancy sequences and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is to derive a formula for the discrepancy of a digital (λ, t,m, s)-net in base b with n = λbm points that requires only O(n) operations to evaluate. The second main result is to derive exact formulas for the gain coefficients of a digital (t,m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, arise in both the worst-case and random-case analyses of quadrature error. In particular, they appear in the formula for the root mean square discrepancy of a scrambled net.