{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T07:33:45Z","timestamp":1785310425800,"version":"3.55.0"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"225","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We provide sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test. For\n                    <inline-formula content-type=\"math\/mathml\">\n                      <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\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>8<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>9<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>64<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>127<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>128<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell = 8, 9, \\dots , 64, 127, 128<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we take as a modulus\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    the largest prime smaller than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 Superscript script l\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">2^\\ell<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and provide a list of multipliers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that the MLCG with modulus\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and multiplier\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has a good lattice structure in dimensions 2 to 32. We provide similar lists for power-of-two moduli\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m equals 2 Superscript script l\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u2113\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m = 2^{\\ell }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , for multiplicative and non-multiplicative LCGs.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-00996-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"249-260","source":"Crossref","is-referenced-by-count":172,"title":["Tables of linear congruential generators of different sizes and good lattice structure"],"prefix":"10.1090","volume":"68","author":[{"given":"Pierre","family":"L\u2019Ecuyer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[1999]]},"reference":[{"key":"1","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2016-7","volume-title":"Sphere packings, lattices and groups","volume":"290","author":"Conway, J. H.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/038796617X"},{"issue":"189","key":"2","doi-asserted-by":"publisher","first-page":"331","DOI":"10.2307\/2008698","article-title":"Multiplicative congruential random number generators with modulus 2^{\ud835\udefd}: an exhaustive analysis for \ud835\udefd=32 and a partial analysis for \ud835\udefd=48","volume":"54","author":"Fishman, George S.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","series-title":"Springer Series in Operations Research","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2553-7","volume-title":"Monte Carlo","author":"Fishman, George S.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/038794527X"},{"key":"4","first-page":"571","article-title":"On a problem of Goursat","volume":"44","author":"B\u01cedescu, Radu","year":"1939","journal-title":"Gaz. Mat."},{"key":"5","series-title":"Addison-Wesley Series in Computer Science and Information Processing","isbn-type":"print","volume-title":"The art of computer programming. Vol. 2","author":"Knuth, Donald E.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0201038226","edition":"2"},{"issue":"6","key":"6","doi-asserted-by":"publisher","first-page":"742","DOI":"10.1145\/62959.62969","article-title":"Efficient and portable combined random number generators","volume":"31","author":"L\u2019Ecuyer, Pierre","year":"1988","journal-title":"Comm. ACM","ISSN":"https:\/\/id.crossref.org\/issn\/0001-0782","issn-type":"print"},{"key":"7","unstructured":"\\bysame, Random number generation, Handbook on Simulation (Jerry Banks, ed.), Wiley, 1998, To appear."},{"key":"8","doi-asserted-by":"crossref","unstructured":"P. L\u2019Ecuyer, F. Blouin, and R. Couture, A search for good multiple recursive random number generators, ACM Transactions on Modeling and Computer Simulation 3 (1993), no. 2, 87\u201398.","DOI":"10.1145\/169702.169698"},{"key":"9","doi-asserted-by":"crossref","unstructured":"P. L\u2019Ecuyer and R. Couture, An implementation of the lattice and spectral tests for multiple recursive linear random number generators, INFORMS Journal on Computing 9 (1997), no. 2, 206\u2013217.","DOI":"10.1287\/ijoc.9.2.206"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1137\/0152056","article-title":"A deterministic particle method for solving kinetic transport equations: the semiconductor Boltzmann equation case","volume":"52","author":"Delaurens, F.","year":"1992","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"M. Sakamoto and S. Morito, Combination of multiplicative congruential random number generators with safe prime modulus, Proceedings of the 1995 Winter Simulation Conference, IEEE Press, 1995, pp. 309\u2013315.","DOI":"10.1109\/WSC.1995.478740"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-225\/S0025-5718-99-00996-5\/S0025-5718-99-00996-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-225\/S0025-5718-99-00996-5\/S0025-5718-99-00996-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:57:51Z","timestamp":1776722271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-225\/S0025-5718-99-00996-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999]]},"references-count":11,"journal-issue":{"issue":"225","published-print":{"date-parts":[[1999,1]]}},"alternative-id":["S0025-5718-99-00996-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-00996-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999]]}}}