{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T14:08:02Z","timestamp":1778767682621,"version":"3.51.4"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,12,27]],"date-time":"2006-12-27T00:00:00Z","timestamp":1167177600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    New lower bounds for three- and four-level designs under the centered\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01806-5","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"859-878","source":"Crossref","is-referenced-by-count":41,"title":["Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels"],"prefix":"10.1090","volume":"75","author":[{"given":"Kai-Tai","family":"Fang","sequence":"first","affiliation":[]},{"given":"Dietmar","family":"Maringer","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Tang","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Winker","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,12,27]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1111\/j.2517-6161.1996.tb02068.x","article-title":"Experimental design and observation for large systems","volume":"58","author":"Bates, R. 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Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"237","key":"6","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1090\/S0025-5718-00-01281-3","article-title":"Centered \ud835\udc3f\u2082-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs","volume":"71","author":"Fang, Kai-Tai","year":"2002","journal-title":"Math. 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Inference","ISSN":"https:\/\/id.crossref.org\/issn\/0378-3758","issn-type":"print"},{"key":"12","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081","volume-title":"Random number generation and quasi-Monte Carlo methods","volume":"63","author":"Niederreiter, Harald","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712955"},{"issue":"6","key":"13","first-page":"485","article-title":"A note on uniform distribution and experimental design","volume":"26","author":"Wang, Yuan","year":"1981","journal-title":"Kexue Tongbao (English Ed.)","ISSN":"https:\/\/id.crossref.org\/issn\/0250-7862","issn-type":"print"},{"key":"14","series-title":"Wiley Series in Probability and Statistics","isbn-type":"print","volume-title":"Optimization heuristics in econometrics","author":"Winker, Peter","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0471856312"},{"key":"15","doi-asserted-by":"crossref","unstructured":"P. Winker and K. T. Fang (1998), Optimal \ud835\udc48-type design, in H. Niederreiter, P. Zinterhof, and P. Hellekalek (eds.), Monte Carlo and Quasi-Monte Carlo Methods 1996, Springer, 436\u2013448.","DOI":"10.1007\/978-1-4612-1690-2_31"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01806-5\/S0025-5718-05-01806-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01806-5\/S0025-5718-05-01806-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:35:20Z","timestamp":1776782120000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01806-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,12,27]]},"references-count":15,"journal-issue":{"issue":"254","published-print":{"date-parts":[[2006,4]]}},"alternative-id":["S0025-5718-05-01806-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01806-5","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":[[2005,12,27]]}}}