{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:00:07Z","timestamp":1776844807239,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"245","license":[{"start":{"date-parts":[[2004,6,3]],"date-time":"2004-06-03T00:00:00Z","timestamp":1086220800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We introduce the class of\n                    <italic>skew-circulant<\/italic>\n                    lattice rules. These are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -dimensional lattice rules that may be generated by the rows of an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s times s\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s \\times s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"delta equals d plus 1 less-than-or-equal-to 47 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b4\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>47<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\delta = d+1 \\leq 47,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we have constructed an infinite sequence of rules\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ModifyingAbove upper Q With caret left-parenthesis 4 comma delta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mover>\n                                  <mml:mi>Q<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">\n                                    ^\n                                    \n                                  <\/mml:mo>\n                                <\/mml:mover>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\hat Q}(4,\\delta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that has a limit rho index of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"27 slash 34 almost-equals 0.79\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>27<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>34<\/mml:mn>\n                            <mml:mo>\n                              \u2248\n                              \n                            <\/mml:mo>\n                            <mml:mn>0.79<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">27\/34 \\approx 0.79<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01534-5","type":"journal-article","created":{"date-parts":[[2003,10,10]],"date-time":"2003-10-10T19:11:14Z","timestamp":1065813074000},"page":"279-295","source":"Crossref","is-referenced-by-count":29,"title":["Four-dimensional lattice rules generated by skew-circulant matrices"],"prefix":"10.1090","volume":"73","author":[{"given":"J.","family":"Lyness","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"S\u00f8revik","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,6,3]]},"reference":[{"issue":"236","key":"1","doi-asserted-by":"publisher","first-page":"1549","DOI":"10.1090\/S0025-5718-01-01326-6","article-title":"Three- and four-dimensional \ud835\udc3e-optimal lattice rules of moderate trigonometric degree","volume":"70","author":"Cools, Ronald","year":"2001","journal-title":"Math. 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V.","year":"1991","journal-title":"Zh. Vychisl. Mat. i Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1090\/psapm\/048\/1314879","article-title":"Computing integrals of the complex error function","author":"Weideman, J. A. C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821802917"},{"key":"10","doi-asserted-by":"crossref","unstructured":"[S{\\o}My01] T. S\u00f8revik and J. F. Myklebust, GRISK: An Internet based search for K-optimal lattice rules, in Proceedings of PARA2000, Lecture Notes in Computer Science 1947, pp. 196\u2013205, Springer Verlag, 2001.","DOI":"10.1007\/3-540-70734-4_24"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2004-73-245\/S0025-5718-03-01534-5\/S0025-5718-03-01534-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-245\/S0025-5718-03-01534-5\/S0025-5718-03-01534-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:38:30Z","timestamp":1776778710000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-245\/S0025-5718-03-01534-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,6,3]]},"references-count":10,"journal-issue":{"issue":"245","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["S0025-5718-03-01534-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01534-5","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":[[2003,6,3]]}}}