Abstract
Sequences of pseudo-random numbers are discussed which are generated by the linear congruential method where the period is equal to the modulus m. Such sequences are divided into non-overlapping vectors with n components. In this way for each initial number exactly m/gcd(n, m) different vectors are obtained. It is shown that the periodic continuation (with period m) of these vectors forms a grid which is a sub-grid of the familiar grid generated by all m overlapping vectors. A sub-lattice structure also exists for certain multiplicative congruential generators which are often used in practice.
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References
Afflerbach, L., Grothe, H.: Calculation of Minkowski-Reduced Lattice Bases. Computing 35, 269–276 (1985)
Beyer, W.A., Roof, R.B., Williamson, D.: The Lattice Structure of Multiplicative Pseudo-Random Vectors. Math. Comput. 25, 345–360 (1971)
Beyer, W.A.: Lattice Structure and Reduced Bases of Random Vectors Generated by Linear Recurrences. In: S.K. Zaremba (Ed.): Applications of Number Theory to Numerical Analysis, 361–370 (1972)
Dieter, U., Ahrens, J.H.: Uniform Random Numbers. Institut f. Math. Stat., Technische Hochschule Graz (1974)
Knuth, D.E.: The Art of Computer Programming, Vol. II, 2d ed. Reading (Mass.), Menlo Park (Cal.), London, Amsterdamm, Don Mills (Ont.), Sydney: Addison-Wesley (1981)
Marsaglia, G.: Random Number Fall Mainly in the Planes. Proc. Nat.Acad.Sci. 61,25–28 (1968)
Marsaglia, G.: Regularities in Congruential Random Number Generators. Nunerische Math. 16, 8–10 (1970)
Marsaglia, G.: The Structure of Linear Congruential Sequences. In: S.K. Zaremba (Ed.): Applications of Number Theory to Numerical Analysis, 249–285 (1972)
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Afflerbach, L. The sub-lattice structure of linear congruential random number generators. Manuscripta Math 55, 455–465 (1986). https://doi.org/10.1007/BF01186658
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DOI: https://doi.org/10.1007/BF01186658