Point sets and sequences with small discrepancy. (English) Zbl 0626.10045

Summary: A systematic theory of a class of point sets called \((t,m,s)\)-nets and of a class of sequences called \((t,s)\)-sequences is developed. On the basis of this theory, point sets and sequences in the s-dimensional unit cube with the smallest discrepancy that is currently known are constructed. Various connections with other areas arise in this work, e.g. with orthogonal Latin squares, finite projective planes, finite fields, and algebraic coding theory. Applications of the theory of \((t,m,s)\)-nets to two methods for pseudorandom number generation, namely the digital multistep method and the GFSR method, are presented. Several open problems, mostly of a combinatorial nature, are stated.


11K06 General theory of distribution modulo \(1\)
05A05 Permutations, words, matrices
05B25 Combinatorial aspects of finite geometries
11T99 Finite fields and commutative rings (number-theoretic aspects)
Full Text: DOI EuDML


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