{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:59:58Z","timestamp":1776844798971,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2002,5,14]],"date-time":"2002-05-14T00:00:00Z","timestamp":1021334400000},"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>A systematic search for optimal lattice rules of specified trigonometric degree<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\"><mml:semantics><mml:mi>d<\/mml:mi><mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>over the hypercube<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 0 comma 1 right-parenthesis Superscript s\"><mml:semantics><mml:mrow><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mi>s<\/mml:mi><\/mml:msup><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">[0,1)^s<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>has been undertaken. The search is restricted to a population<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K left-parenthesis s comma delta right-parenthesis\"><mml:semantics><mml:mrow><mml:mi>K<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03b4<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">K(s,\\delta )<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>of lattice rules<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q left-parenthesis normal upper Lamda right-parenthesis\"><mml:semantics><mml:mrow><mml:mi>Q<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi mathvariant=\"normal\">\u039b<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">Q(\\Lambda )<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>. This includes those where the dual lattice<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Lamda Superscript up-tack\"><mml:semantics><mml:msup><mml:mi mathvariant=\"normal\">\u039b<\/mml:mi><mml:mo>\u22a5<\/mml:mo><\/mml:msup><mml:annotation encoding=\"application\/x-tex\">\\Lambda ^\\perp<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>may be generated by<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\"><mml:semantics><mml:mi>s<\/mml:mi><mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>points<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold h\"><mml:semantics><mml:mi mathvariant=\"bold\">h<\/mml:mi><mml:annotation encoding=\"application\/x-tex\">\\bf h<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>for each of which<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue bold h EndAbsoluteValue equals delta equals d plus 1\"><mml:semantics><mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|<\/mml:mo><\/mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"bold\">h<\/mml:mtext><\/mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mi>\u03b4<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>d<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">|\\textbf {h} | = \\delta =d+1<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s equals 3\"><mml:semantics><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">s=3<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>,<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d less-than-or-equal-to 29\"><mml:semantics><mml:mrow><mml:mi>d<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mn>29<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">d \\leq 29<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>and<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s equals 4\"><mml:semantics><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">s=4<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>,<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d less-than-or-equal-to 23\"><mml:semantics><mml:mrow><mml:mi>d<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mn>23<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">d \\leq 23<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>, a list of<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\"><mml:semantics><mml:mi>K<\/mml:mi><mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>-optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.<\/p>","DOI":"10.1090\/s0025-5718-01-01326-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1549-1567","source":"Crossref","is-referenced-by-count":22,"title":["Three- and four-dimensional \ud835\udc3e-optimal lattice rules of moderate trigonometric degree"],"prefix":"10.1090","volume":"70","author":[{"given":"Ronald","family":"Cools","sequence":"first","affiliation":[]},{"given":"James","family":"Lyness","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,5,14]]},"reference":[{"key":"1","first-page":"13","article-title":"A relation between cubature formulae of trigonometric degree and lattice rules","author":"Beckers, Marc","year":"1993"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/s006070050018","article-title":"Smolyak\u2019s construction of cubature formulas of arbitrary trigonometric degree","volume":"62","author":"Cools, R.","year":"1999","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jcom.1997.0443","article-title":"Different quality indexes for lattice rules","volume":"13","author":"Cools, Ronald","year":"1997","journal-title":"J. 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Noskov, On the construction of cubature formulae of higher trigonometric degree, Metody Vy\u010disl. 16 (1991), 16\u201323 (Russian)."},{"issue":"10","key":"20","first-page":"5","article-title":"Cubature formulas of increased trigonometric accuracy for periodic functions of four variables","volume":"36","author":"Noskov, M. V.","year":"1996","journal-title":"Zh. Vychisl. Mat. i Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"key":"21","unstructured":"[Sem96] A. R. Semenova, Computing experiments for construction of cubature formulae of high trigonometric accuracy, Cubature Formulas and Their Applications (Russian) (Ufa) (M. D. Ramazanov, ed.), 1996, pp. 105\u2013115."},{"key":"22","series-title":"Oxford Science Publications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534723.001.0001","volume-title":"Lattice methods for multiple integration","author":"Sloan, I. 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