Low-discrepancy point sets obtained by digital constructions over finite fields. (English) Zbl 0757.11024

In the paper under review, low-discrepancy point sets in \([0,1)^ s\) are constructed in which the coordinates of the points are given by digit expansions in a base \(q^ m\), \(q\) prime. These point sets are \((t,m,s)\)- nets in the sense of the author [Monatsh. Math. 104, 273-337 (1987; Zbl 0626.10045)]. Thus upper and lower bounds for the star discrepancy \(D^*_ N\) of the constructed point sets can be established. An average upper bound is \(D^*_ N=O(N^{-1}(\log N)^ s)\), where \(N=q^ m\).
For the efficient implementation of the point sets, a further specialization is investigated where the digit expansion is derived from Laurent series expansions. Then the point sets can be calculated by linear recurrence relations as in [the author, J. Number Theory 30, 51-70 (1988; Zbl 0651.10034)]. A two-dimensional point set with \(D^*_ N=O(N^{-1}\log N)\) is also constructed. The proofs depend among other things on the theory of arithmetic functions on a Galois field as developed by Carlitz.


11K38 Irregularities of distribution, discrepancy
65C05 Monte Carlo methods
65D30 Numerical integration
Full Text: DOI EuDML


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