{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:35:36Z","timestamp":1778495736036,"version":"3.51.4"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"218","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We study the multiply-with-carry family of generators proposed by Marsaglia as a generalization of previous add-with-carry families. We define for them an infinite state space and focus our attention on the (finite) subset of recurrent states. This subset will, in turn, split into possibly several subgenerators. We discuss the uniformity of the\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -dimensional distribution of the output of these subgenerators over their full period. In order to improve this uniformity for higher dimensions, we propose a method for finding good parameters in terms of the spectral test. Our results are stated in a general context and are applied to a related complementary multiply-with-carry family of generators.\n <\/p>","DOI":"10.1090\/s0025-5718-97-00827-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"591-607","source":"Crossref","is-referenced-by-count":28,"title":["Distribution properties of multiply-with-c arry random number generators"],"prefix":"10.1090","volume":"66","author":[{"given":"Raymond","family":"Couture","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre","family":"L\u2019Ecuyer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"R. Couture and P. L\u2019Ecuyer, On the lattice structure of certain linear congruential sequences related to AWC\/SWB generators, Mathematics of Computation 62 (1994), no. 206, 798-808. \\MR{94g:65007}","DOI":"10.1090\/S0025-5718-1994-1220826-X"},{"key":"2","doi-asserted-by":"crossref","unstructured":"E. Hecke, Lectures on the theory of algebraic numbers, Springer-Verlag, New York, 1981. \\MR{83m:12001}","DOI":"10.1007\/978-1-4757-4092-9"},{"key":"3","unstructured":"D. E. Knuth, The art of computer programming, volume 2: Seminumerical algorithms, second ed., Addison-Wesley, Reading, Mass., 1981. \\MR{83i:68003}"},{"key":"4","doi-asserted-by":"crossref","unstructured":"P. L\u2019Ecuyer, Uniform random number generation, Annals of Operations Research 53 (1994), 77-120. \\MR{95k:65007}","DOI":"10.1007\/BF02136827"},{"key":"5","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, Informes Journal on Computing, to appear, 1997.","DOI":"10.1287\/ijoc.9.2.206"},{"key":"6","unstructured":"G. Marsaglia, Yet another rng, Posted to the electronic billboard sci.stat.math, August 1, 1994."},{"key":"7","doi-asserted-by":"crossref","unstructured":"G. Marsaglia and A. Zaman, A new class of random number generators, The Annals of Applied Probability 1 (1991), 462-480. \\MR{92h:65009}","DOI":"10.1214\/aoap\/1177005878"},{"key":"8","unstructured":"W. Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN-Polish Scientific Publishers, Warsaw, 1974. \\MR{50:268}"},{"key":"9","doi-asserted-by":"crossref","unstructured":"H. Niederreiter, Random number generation and quasi-Monte Carlo methods, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, SIAM, Philadelphia, 1992. \\MR{93h:65008}","DOI":"10.1137\/1.9781611970081"},{"key":"10","doi-asserted-by":"crossref","unstructured":"C. L. Siegel, \u00dcber die Classenzahl quadratischer Zahlk\u00f6rper, Acta Arithmetica 1 (1935), 83-86.","DOI":"10.4064\/aa-1-1-83-86"},{"key":"11","doi-asserted-by":"crossref","unstructured":"S. Tezuka, P. L\u2019Ecuyer, and R. Couture, On the add-with-carry and subtract-with-borrow random number generators, ACM Transactions of Modeling and Computer Simulation 3 (1994), no. 4, 315-331.","DOI":"10.1145\/159737.159749"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00827-2\/S0025-5718-97-00827-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00827-2\/S0025-5718-97-00827-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:26:50Z","timestamp":1776720410000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00827-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"references-count":11,"journal-issue":{"issue":"218","published-print":{"date-parts":[[1997,4]]}},"alternative-id":["S0025-5718-97-00827-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-97-00827-2","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":[[1997]]}}}