{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T01:10:30Z","timestamp":1785287430827,"version":"3.55.0"},"reference-count":37,"publisher":"American Mathematical Society (AMS)","issue":"298","license":[{"start":{"date-parts":[[2016,7,14]],"date-time":"2016-07-14T00:00:00Z","timestamp":1468454400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["FT110100065"],"award-info":[{"award-number":["FT110100065"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We report the results of a computer investigation of sets of mutually orthogonal Latin squares (MOLS) of small order. For\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n less-than-or-slanted-equals 9\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2a7d\n                              \n                            <\/mml:mo>\n                            <mml:mn>9<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\leqslant 9<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we:\n                  <\/p>\n                  <p>\n                    determine the number of orthogonal mates for each species of Latin square of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ;\n                  <\/p>\n                  <p>\n                    calculate the proportion of Latin squares of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that have an orthogonal mate, and the expected number of mates when a square is chosen uniformly at random;\n                  <\/p>\n                  <p>\n                    classify all sets of MOLS of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    up to various different notions of equivalence.\n                  <\/p>\n                  <p>We also provide a triple of Latin squares of order 10 that is the closest to being a set of MOLS so far found.<\/p>","DOI":"10.1090\/mcom\/3010","type":"journal-article","created":{"date-parts":[[2015,7,14]],"date-time":"2015-07-14T11:56:26Z","timestamp":1436874986000},"page":"799-824","source":"Crossref","is-referenced-by-count":35,"title":["Enumeration of MOLS of small order"],"prefix":"10.1090","volume":"85","author":[{"given":"Judith","family":"Egan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ian","family":"Wanless","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,7,14]]},"reference":[{"key":"1","unstructured":"R. C. Bose, On the application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares, Sankhy\u0101 3 (1938), 323\u2013338."},{"issue":"1-2","key":"2","first-page":"113","article-title":"A pair of orthogonal Latin squares of order 10 with four shared parallel transversals","volume":"18","author":"Brown, J. W.","year":"1993","journal-title":"J. Combin. Inform. System Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0250-9628","issn-type":"print"},{"key":"3","series-title":"Discrete Mathematics and its Applications (Boca Raton)","isbn-type":"print","volume-title":"Handbook of combinatorial designs","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9781584885061","edition":"2"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"796","DOI":"10.1016\/j.jcta.2010.11.011","article-title":"Monogamous Latin squares","volume":"118","author":"Danziger, Peter","year":"2011","journal-title":"J. Combin. Theory Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0097-3165","issn-type":"print"},{"key":"5","unstructured":"J. D\u00e9nes and A.D. Keedwell, Latin Squares and their Applications. Akad\u00e9miai Kiad\u00f3, Budapest, 1974."},{"issue":"7","key":"6","doi-asserted-by":"publisher","first-page":"1562","DOI":"10.1016\/j.jcta.2013.05.004","article-title":"On the number of transversal designs","volume":"120","author":"Donovan, D. M.","year":"2013","journal-title":"J. Combin. Theory Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0097-3165","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/0378-3758(77)90019-2","article-title":"Maximal sets of Latin squares and partial transversals","volume":"1","author":"Drake, David A.","year":"1977","journal-title":"J. Statist. Plann. Inference","ISSN":"https:\/\/id.crossref.org\/issn\/0378-3758","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1023\/B:DESI.0000032607.03637.1c","article-title":"The non-existence of maximal sets of four mutually orthogonal Latin squares of order 8","volume":"33","author":"Drake, David A.","year":"2004","journal-title":"Des. Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1016\/j.jspi.2010.06.020","article-title":"Indivisible partitions of Latin squares","volume":"141","author":"Egan, Judith","year":"2011","journal-title":"J. Statist. Plann. Inference","ISSN":"https:\/\/id.crossref.org\/issn\/0378-3758","issn-type":"print"},{"issue":"7","key":"10","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1002\/jcd.21316","article-title":"Erratum: Latin squares with restricted transversals [MR2868854]","volume":"20","year":"2012","journal-title":"J. Combin. Des.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-8539","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s10623-006-8153-3","article-title":"Latin squares without orthogonal mates","volume":"40","author":"Evans, Anthony B.","year":"2006","journal-title":"Des. Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"},{"key":"12","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1093\/biomet\/57.2.351","article-title":"The existence and construction of two families of designs for two successive experiments","volume":"57","author":"Hedayat, A.","year":"1970","journal-title":"Biometrika","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3444","issn-type":"print"},{"key":"13","unstructured":"K. Hicks, G. L. Mullen, L. Storme and J. Vanpoucke, The number of different reduced complete sets of mutually orthogonal Latin squares corresponding to the Desarguesian projective planes, submitted for publication."},{"issue":"274","key":"14","doi-asserted-by":"publisher","first-page":"1197","DOI":"10.1090\/S0025-5718-2010-02420-2","article-title":"The number of Latin squares of order 11","volume":"80","author":"Hulpke, Alexander","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0012-365X(86)90163-9","article-title":"Maximal difference matrices of order \u226410","volume":"58","author":"Jungnickel, Dieter","year":"1986","journal-title":"Discrete Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-365X","issn-type":"print"},{"key":"16","series-title":"Algorithms and Computation in Mathematics","isbn-type":"print","volume-title":"Classification algorithms for codes and designs","volume":"15","author":"Kaski, Petteri","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540289906"},{"key":"17","unstructured":"P. Kaski and O. Pottonen, \\url{http:\/\/koti.kapsi.fi\/pottonen\/libexact.html} libexact software."},{"key":"18","doi-asserted-by":"crossref","unstructured":"J. I. Kokkala and P. R.J. \u00d6sterg\u00e5rd, Classification of Graeco-Latin cubes, to appear J. Combin. Des., DOI: 10.1002\/jcd.21400.","DOI":"10.1002\/jcd.21400"},{"issue":"6","key":"19","doi-asserted-by":"publisher","first-page":"2889","DOI":"10.1090\/S0002-9947-00-02350-3","article-title":"The structure of conjugacy closed loops","volume":"352","author":"Kunen, Kenneth","year":"2000","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"20","series-title":"Wiley-Interscience Series in Discrete Mathematics and Optimization","isbn-type":"print","volume-title":"Discrete mathematics using Latin squares","author":"Laywine, Charles F.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0471240648"},{"issue":"1","key":"21","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1002\/jcd.10064","article-title":"Atomic Latin squares of order eleven","volume":"12","author":"Maenhaut, Barbara M.","year":"2004","journal-title":"J. Combin. Des.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-8539","issn-type":"print"},{"key":"22","unstructured":"B.D. McKay, Latin squares, \\url{http:\/\/cs.anu.edu.au\/ bdm\/data\/Latin.html}."},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/s10623-006-0012-8","article-title":"The number of transversals in a Latin square","volume":"40","author":"McKay, Brendan D.","year":"2006","journal-title":"Des. Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"},{"issue":"2","key":"24","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1002\/jcd.20105","article-title":"Small Latin squares, quasigroups, and loops","volume":"15","author":"McKay, Brendan D.","year":"2007","journal-title":"J. Combin. Des.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-8539","issn-type":"print"},{"key":"25","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.jsc.2013.09.003","article-title":"Practical graph isomorphism, II","volume":"60","author":"McKay, Brendan D.","year":"2014","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"3","key":"26","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/s00026-005-0261-7","article-title":"On the number of Latin squares","volume":"9","author":"McKay, Brendan D.","year":"2005","journal-title":"Ann. Comb.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-0006","issn-type":"print"},{"issue":"3","key":"27","doi-asserted-by":"publisher","first-page":"264","DOI":"10.2307\/2369797","article-title":"Tactical Memoranda I\u2013III","volume":"18","author":"Moore, Eliakim Hastings","year":"1896","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"28","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1111\/j.1469-1809.1939.tb02214.x","article-title":"The 7 x 7 squares","volume":"9","author":"Norton, H. W.","year":"1939","journal-title":"Ann. Eugenics","ISSN":"https:\/\/id.crossref.org\/issn\/2050-1420","issn-type":"print"},{"key":"29","isbn-type":"print","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1017\/CBO9780511734885.004","article-title":"Constructing combinatorial objects via cliques","author":"\u00d6sterg\u00e5rd, Patric R. J.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521615232"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1016\/0097-3165(92)90067-5","article-title":"Complete sets of pairwise orthogonal Latin squares and the corresponding projective planes","volume":"59","author":"Owens, P. J.","year":"1992","journal-title":"J. Combin. Theory Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0097-3165","issn-type":"print"},{"key":"31","first-page":"83","article-title":"Complete sets of pairwise orthogonal Latin squares of order 9","volume":"18","author":"Owens, P. J.","year":"1995","journal-title":"J. Combin. Math. Combin. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0835-3026","issn-type":"print"},{"key":"32","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/S0012-365X(97)89938-4","article-title":"Aspects of complete sets of 9\u00d79 pairwise orthogonal Latin squares","volume":"167\/168","author":"Owens, P. J.","year":"1997","journal-title":"Discrete Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-365X","issn-type":"print"},{"key":"33","first-page":"193","article-title":"Subsquares and transversals in Latin squares","volume":"29","author":"van Rees, G. H. J.","year":"1990","journal-title":"Ars Combin.","ISSN":"https:\/\/id.crossref.org\/issn\/0381-7032","issn-type":"print"},{"key":"34","doi-asserted-by":"publisher","first-page":"Research Paper 9, 16","DOI":"10.37236\/1441","article-title":"Perfect factorisations of bipartite graphs and Latin squares without proper subrectangles","volume":"6","author":"Wanless, I. M.","year":"1999","journal-title":"Electron. J. Combin."},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"Research Paper 12, 15","DOI":"10.37236\/1629","article-title":"A generalisation of transversals for Latin squares","volume":"9","author":"Wanless, Ian M.","year":"2002","journal-title":"Electron. J. Combin."},{"key":"36","unstructured":"I.M. Wanless, \\url{http:\/\/users.monash.edu.au\/ iwanless\/data\/MOLS\/}, author\u2019s homepage."},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s10623-006-8168-9","article-title":"The existence of Latin squares without orthogonal mates","volume":"40","author":"Wanless, Ian M.","year":"2006","journal-title":"Des. Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-03010-5\/S0025-5718-2015-03010-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-03010-5\/S0025-5718-2015-03010-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:42:12Z","timestamp":1776796932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-03010-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,14]]},"references-count":37,"journal-issue":{"issue":"298","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["S0025-5718-2015-03010-5"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3010","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":[[2015,7,14]]}}}