Gottlieb Pirsic

A Software Implementation of Niederreiter-Xing Sequences

In a series of papers, Niederreiter and Xing introduced new construction methods for low-discrepancy sequences, more specifically (t,s)-sequences. As these involve the rather abstract theory of algebraic function fields — a special case of algebraic geometry and also closely related to function theory and algebraic number theory — for a long time no computer implementation of this new method was given. In this paper we present our efforts in this direction, address the algorithmical problems and give some numerical data obtained from our implementation.

Calculation of the Quality Parameter of Digital Nets and Application to Their Construction

In quasi-Monte Carlo methods, point sets of low discrepancy are crucial for accurate results. A class of point sets with low theoretic upper bounds of discrepancy are the digital point sets known as digital (t, m, s)-nets which can be implemented very efficiently. The parameter t is indicative of the quality; i.e., small values of t lead to small upper bounds of the discrepancy. We introduce an effective way to establish this quality parameter t for digital nets constructed over arbitrary finite fields and give an application to the construction of digital nets of high quality.

A Kronecker Product Construction for Digital Nets

An analogy of (t,m, s)-nets with codes, viz. the notion of the dual space of a digital net, is used to obtain a new way of constructing digital nets. The method is reminiscent of the Kronecker product code construction in the theory of linear codes.

On the structure of digital explicit nonlinear and inversive pseudorandom number generators

We analyze the lattice structure and distribution of the digital explicit inversive pseudorandom number generator introduced by Niederreiter and Winterhof as well as of a general digital explicit nonlinear generator. In particular, we extend a lattice test designed for this class of pseudorandom number generators to parts of the period and arbitrary lags and prove that these generators pass this test up to very high dimensions. We also analyze the behavior of digital explicit inversive and nonlinear generators under another very strong lattice test which in its easiest form can be traced back to Marsaglia and provides a complexity measure essentially equivalent to linear complexity.

The Microstructure of (t, m, s)-Nets

In an article of A. B. Owen (1998, J. Complexity14, 466?489) the question about the distribution properties of digital (t, m, s)-nets in small intervals was raised. We give upper and lower bounds for the maximum number of points of a (t, m, s)-net in these intervals and also provide a way of improving the distribution properties in some cases.

Discrepancy of Hyperplane Nets and Cyclic Nets

Digital nets are very important representatives in the family of low-discrepancy point sets which are often used as underlying nodes for quasi-Monte Carlo integration rules. Here we consider a special sub-class of digital nets known as cyclic nets and, more general, hyperplane nets. We show the existence of such digital nets of good quality with respect to star discrepancy in the classical as well as weighted case and we present effective search algorithms based on a component-by-component construction.

Extensible hyperplane nets

Abstract Extensible (polynomial) lattice point sets have the property that the number N of points in the node set of a quasi-Monte Carlo algorithm may be increased while retaining the existing points. Explicit constructions for extensible (polynomial) lattice point sets have been presented recently by Niederreiter and Pillichshammer. It is the aim of this paper to establish extensibility for a powerful generalization of polynomial lattice point sets, the so-called hyperplane nets.

On the existence and distribution quality of hyperplane sequences

Abstract It is well known that digital ( t , m , s ) -nets and ( T , s ) -sequences over a finite field have excellent properties when they are used as underlying nodes in quasi-Monte Carlo integration rules. One very general sub-class of digital nets are hyperplane nets which can be viewed as a generalization of cyclic nets and of polynomial lattice point sets. In this paper we introduce infinite versions of hyperplane nets and call these sequences hyperplane sequences. Our construction is based on the recent duality theory for digital sequences according to Dick and Niederreiter. We then analyze the equidistribution properties of hyperplane sequences in terms of the quality function T and the star discrepancy.