The discrepancy and gain coefficients of scrambled digital nets

Rong Xian Yue*, Fred J. Hickernell

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

21 Citations (Scopus)


Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets 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 a formula for the discrepancy of a scrambled 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 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, determine both the worst-case and random-case analyses of quadrature error.

Original languageEnglish
Pages (from-to)135-151
Number of pages17
JournalJournal of Complexity
Issue number1
Publication statusPublished - Mar 2002
Externally publishedYes

Scopus Subject Areas

  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Mathematics(all)
  • Control and Optimization
  • Applied Mathematics


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