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10 - Controlling the shape of generating matrices in global function field constructions of digital sequences
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- By Roswitha Hofer, Johannes Kepler University Linz, Isabel Pirsic, Johannes Kepler University Linz
- Edited by Gerhard Larcher, Johannes Kepler Universität Linz, Friedrich Pillichshammer, Johannes Kepler Universität Linz, Arne Winterhof, Chaoping Xing, Nanyang Technological University, Singapore
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- Book:
- Applied Algebra and Number Theory
- Published online:
- 18 December 2014
- Print publication:
- 11 December 2014, pp 164-189
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- Chapter
- Export citation
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Summary
This paper is dedicated to H. Niederreiter on the occasion of his 70th birthday.
Abstract
Motivated by computational as well as theoretical considerations, we show how the shape and density of the generating matrices of two optimal constructions of (t, s)-sequences and (u, e, s)-sequences (the Xing–Niederreiter and Hofer–Niederreiter sequences) can be controlled by a careful choice of various parameters. We also present some experimental data to support our assertions and point out open problems.
Introduction
The usefulness of and need for well-distributed pseudorandom and quasi-random point sets in very high dimensions has been evidenced by the unbroken stream of publications and conferences with the topic of Monte Carlo and quasi-Monte Carlo (MCQMC) methods in scientific computing, most notably the biannual conference series and proceedings of the same name. Beginning with the well-known Koksma–Hlawka inequality up to the more recent higher order nets, it became clear that, in particular, applications pertaining to multivariate numerical integration are an important area covered by MCQMC methods. Numerous applications in diverse areas of applied mathematics profit from this fact; often cited are applications in finance, computer aided visualization and simulations. (The reader is referred to [5], [4], and [18].)
As regards the suitability of even arbitrary point sets for MCQMC methods, the notion of discrepancy is well established as a measure for the degree of equidistribution, which significantly determines, for example, the error of numerical integration. In brief, discrepancy can be defined as measuring the worst case integration error when applied to indicator functions of subintervals of the unit cube. When the coordinates of the intervals are restricted to b-adic rationals, we arrive at the notion of (t, s)-sequences (in base b)[16]; if, furthermore, a different granularity is permitted in different coordinates, we arrive at the recent refinement of (u, e, s)-sequences [10, 26].
We review the definitions of these concepts in more detail. In the following, let b ∈ ℕ {1}; N, m, s ∈ ℕ t, u ∈ ℕ0 and q ∈ ℕ a prime power.