### Abstract

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F^{2} The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the fc-dimensional hypercube into 2^{kl}identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n . We give numerical examples and discuss the practical implications of our results.

Original language | English |
---|---|

Pages (from-to) | 749-761 |

Number of pages | 13 |

Journal | Mathematics of Computation |

Volume | 60 |

Issue number | 202 |

DOIs | |

Publication status | Published - Apr 1993 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*60*(202), 749-761. https://doi.org/10.1090/S0025-5718-1993-1176708-4

**On the distribution of fc-dimensional vectors for simple and combined tausworthe sequences.** / Couture, Raymond; Lecuyer, Pierre; Tezuka, Shu.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 60, no. 202, pp. 749-761. https://doi.org/10.1090/S0025-5718-1993-1176708-4

}

TY - JOUR

T1 - On the distribution of fc-dimensional vectors for simple and combined tausworthe sequences

AU - Couture, Raymond

AU - Lecuyer, Pierre

AU - Tezuka, Shu

PY - 1993/4

Y1 - 1993/4

N2 - The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F2 The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the fc-dimensional hypercube into 2klidentical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n . We give numerical examples and discuss the practical implications of our results.

AB - The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F2 The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the fc-dimensional hypercube into 2klidentical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n . We give numerical examples and discuss the practical implications of our results.

UR - http://www.scopus.com/inward/record.url?scp=84968491617&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968491617&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1993-1176708-4

DO - 10.1090/S0025-5718-1993-1176708-4

M3 - Article

AN - SCOPUS:84968491617

VL - 60

SP - 749

EP - 761

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 202

ER -