Introduction

The explicit construction of point sets which fill out the -dimensional unit cube as uniformly as possible is an important task for many numerical applications. Pseudo-random numbers are often used for this purpose. However, it is well-known that pseudo-random numbers are only a subsitute for true random numbers and tend to show clustering effects. This led to the search for more evenly spread sequences. The general study of uniformly distributed sequences was initiated by Herman Weyl in 1916 with a famous paper "Über die Gleichverteilung von Zahlen mod. Eins", W16. He defined the notion of discrepancy to quantify the quality of uniformity of a finite point set. Now, a sequence is called a low-discrepancy sequence or quasi-random sequence if the discrepancy of the first points decays asymptotically as . There are various explicit constructions, based on combinatiorial methods, algebraic number theory and ergodic theory.
The construction and application of low-discrepancy sequences recieved a lot of attention over the last decades in various fields of numerical computation, as well as in theoretical mathematics. For example, Niederreiter and Xing used algebraic curves over finite fields to construct low-discrepancy sequences with excellent uniform behaviour and the best currently known parameters.
We provide a Mathematica demonstration package which can be used as a starting point to explore some of the concepts and ideas in the area of low-discrepancy sequences. All the calculation can  be done at full precision and in case of the Niederreiter sequence even formal calculation in any finite field is possible. The focus of this package is in demonstration and presentation of the basic concepts and not in speed. A complete set of optimized Mathematica packages, based on an object oriented stream design, together with extensive documentation is available at http://www.mathdirect.com. To boost the performance for demanding applications, a C++ template implementation is also available, which is included into Mathematica via MathLink and runs on Windows NT/9x and various Unix platforms (Solaris, Irix, AIX, Linux, others on request).

Converted by Mathematica November 18, 1998