Tables of linear congruential generators of different sizes and good lattice structure 论文

摘要

We provide sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l equals 8 comma 9 comma ellipsis comma 64 comma 127 comma 128"> <mml:semantics> <mml:mrow> <mml:mi> ℓ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> <mml:mo>,</mml:mo> <mml:mn>9</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mn>64</mml:mn> <mml:mo>,</mml:mo> <mml:mn>127</mml:mn> <mml:mo>,</mml:mo> <mml:mn>128</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell = 8, 9, \dots , 64, 127, 128</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we take as a modulus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the largest prime smaller than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript script l"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mi> ℓ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">2^\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and provide a list of multipliers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the MLCG with modulus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and multiplier <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a good lattice structure in dimensions 2 to 32. We provide similar lists for power-of-two moduli <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 2 Superscript script l"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ℓ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">m = 2^{\ell }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for multiplicative and non-multiplicative LCGs.