{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T06:30:08Z","timestamp":1770964208715,"version":"3.50.1"},"reference-count":93,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T00:00:00Z","timestamp":1734480000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","award":["RF-2024-116"],"award-info":[{"award-number":["RF-2024-116"]}],"id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","award":["EP\/V519686\/1"],"award-info":[{"award-number":["EP\/V519686\/1"]}],"id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC Industrial Case","doi-asserted-by":"publisher","award":["RF-2024-116"],"award-info":[{"award-number":["RF-2024-116"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC Industrial Case","doi-asserted-by":"publisher","award":["EP\/V519686\/1"],"award-info":[{"award-number":["EP\/V519686\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Latin squares are an essential tool in the construction of combinatorial designs. Optimal solutions for problems such as scheduling problems and permutation arrays for powerline communication rely on the ability to construct sets of mutually orthogonal Latin squares (MOLS) that are as large as possible. Although constructions of suitable sets are known, they are scattered among a wide variety of sources, and can be both difficult to understand and contain errors. We describe our experience implementing the largest known sets of MOLS of order n, for n up to 500. We give a source for each construction, provide additional hints for the difficult cases, and correct some errors along the way. We also give constructions for new sets of MOLS of order n, where n is 486, 567, 622, 635, 754, 756, 764, 766, 774, 778, 802, 810, 822, 826, 894, 906, 916, 920 or 936.<\/jats:p>","DOI":"10.3390\/sym16121678","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T09:43:03Z","timestamp":1734514983000},"page":"1678","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Implementing the MOLS Table for n Up to 500"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0941-1717","authenticated-orcid":false,"given":"Alice","family":"Miller","sequence":"first","affiliation":[{"name":"School of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3632-9612","authenticated-orcid":false,"given":"R. Julian R.","family":"Abel","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1116-875X","authenticated-orcid":false,"given":"Ivaylo","family":"Valkov","sequence":"additional","affiliation":[{"name":"School of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5291-6519","authenticated-orcid":false,"given":"Douglas","family":"Fraser","sequence":"additional","affiliation":[{"name":"School of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Colbourn, C.J., Dinitz, J.H., and Stinson, D.R. (1999). Applications of Combinatorial Designs to Communications, Cryptography, and Networking, Cambridge University Press.","DOI":"10.1017\/CBO9780511721335.004"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1109\/JLT.2003.816819","article-title":"Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs","volume":"21","author":"Djordjevic","year":"2003","journal-title":"J. Light. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s00362-018-01071-x","article-title":"Multi-part balanced incomplete-block designs","volume":"60","author":"Bailey","year":"2019","journal-title":"Stat. Pap."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s00453-013-9791-2","article-title":"Efficient Two-Stage Group Testing Algorithms for Genetic Screening","volume":"67","author":"Huber","year":"2013","journal-title":"Algorithmica"},{"key":"ref_5","unstructured":"Keedwell, A.D., and D\u00e9nes, J. (2015). Latin Squares and Their Applications, Elsevier Science."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Colbourn, C.J., and Dinitz, J.H. (2007). Handbook of Combinatorial Designs, Chapman and Hall\/CRC. [2nd ed.].","DOI":"10.1201\/9781420010541"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/0012-365X(75)90040-0","article-title":"Balanced incomplete block designs and related designs","volume":"11","author":"Hanani","year":"1975","journal-title":"Discret. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1002\/jcd.3180030508","article-title":"Some new BIBDs with k=6 and \u03bb=1","volume":"3","author":"Abel","year":"1995","journal-title":"J. Comb. Des."},{"key":"ref_9","first-page":"177","article-title":"Some new RBIBDs with block size 5 and PBDs with block sizes \u22611(mod 5)","volume":"15","author":"Abel","year":"1997","journal-title":"Australas. J. Comb."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1002\/jcd.20157","article-title":"Pair Covering and Other Designs with Block Size 6","volume":"15","author":"Abel","year":"2007","journal-title":"J. Comb. Des."},{"key":"ref_11","first-page":"3","article-title":"Recursive Constructions of Balanced Incomplete Block Designs with Block Size of 7, 8 or 9","volume":"60","author":"Greig","year":"2001","journal-title":"Ars Comb."},{"key":"ref_12","unstructured":"Abel, R.J.R., and Greig, M. (2007). BIBDs with small block size. Handbook of Combinatorial Designs, CRC."},{"key":"ref_13","first-page":"33","article-title":"Some group divisible design constructions","volume":"27","author":"Greig","year":"1998","journal-title":"JCMCC J. Comb. Math. Comb. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1002\/(SICI)1520-6610(1999)7:5<341::AID-JCD5>3.0.CO;2-1","article-title":"Designs from Projective Planes and PBD bases","volume":"7","author":"Greig","year":"1999","journal-title":"J. Comb. Des."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1002\/1520-6610(2000)8:5<363::AID-JCD6>3.0.CO;2-C","article-title":"Group-divisible designs with block size k having k+1 groups, for k=4, 5","volume":"8","author":"Rees","year":"2000","journal-title":"J. Comb. Des."},{"key":"ref_16","first-page":"93","article-title":"Some matrix constructions of L2-type Latin square designs","volume":"95","author":"Saurabh","year":"2022","journal-title":"Bull. Inst. Comb. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wallis, W.D., Street, A.P., and Wallis, J.S. (1972). Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Springer.","DOI":"10.1007\/BFb0069907"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/S0167-5060(08)70710-6","article-title":"Some results on the existence of squares","volume":"6","author":"Mullin","year":"1980","journal-title":"Ann. Discret. Math."},{"key":"ref_19","first-page":"65","article-title":"Designing tournaments with the aid of Latin squares: A presentation of old and new results","volume":"58","author":"Keedwell","year":"2000","journal-title":"Util. Math."},{"key":"ref_20","unstructured":"Bao, L. (November, January 31). MASL: Multiple Access Scheduling Based on Latin Squares. Proceedings of the IEEE MILCOM 2004, Military Communications Conference, Monterey, CA, USA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1080\/09720529.2010.10698290","article-title":"Design of strong cryptographic schemes based on Latin Squares","volume":"13","author":"Pal","year":"2010","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_22","first-page":"191","article-title":"On a Problem in Combinations","volume":"2","author":"Kirkman","year":"1847","journal-title":"Camb. Dublin Math. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1090\/pspum\/019\/9959","article-title":"Solution of Kirkman\u2019s school girl problem","volume":"19","author":"Wilson","year":"1971","journal-title":"Proc. Symp. Pure Math."},{"key":"ref_24","unstructured":"Triska, M. (2008). Solution Methods for the Social Golfer Problem. [Master\u2019s Thesis, Technische Universit\u00e4t Wien]."},{"key":"ref_25","unstructured":"Harvey, W. (2024, November 07). CSPLib Problem 010: Social Golfers Problem. Available online: http:\/\/www.csplib.org\/Problems\/prob010."},{"key":"ref_26","unstructured":"Pegg, E. (2024, November 07). Math Games: Social Golfer Problem. Available online: http:\/\/www.mathpuzzle.com\/MAA\/54-Golf%20Tournaments\/mathgames_08_14_07.html."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Miller, A., Barr, M., Kavanagh, W., Valkov, I., and Purchase, H.C. (2021). Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes. Symmetry, 13.","DOI":"10.3390\/sym13010013"},{"key":"ref_28","first-page":"68","article-title":"Existence of resolvable group divisible designs with block size four and group size two or three","volume":"1","author":"Shen","year":"1996","journal-title":"J. Shanghai Jiaotong Univ. (Engl. Ed.)"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1002\/(SICI)1520-6610(1999)7:5<375::AID-JCD6>3.0.CO;2-K","article-title":"A Steiner 2-Design with an automorphism fixing exactly r+2 points","volume":"7","author":"Colbourn","year":"1998","journal-title":"J. Comb. Des."},{"key":"ref_30","unstructured":"Aguado, A. (2024, December 12). A 10 Days Solution to the Social Golfer Problem. Available online: https:\/\/www.mathpuzzle.com\/MAA\/54-Golf%20Tournaments\/socgolf1.pdf."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Liu, K., L\u00f6ffler, S., and Hofstedt, P. (2019, January 19\u201321). Solving the Social Golfers Problems by Constraint Programming in Sequential and Parallel. Proceedings of the 11th International Conference on Agents and Artificial Intelligence (ICAART 2019), Prague, Czech Republic.","DOI":"10.5220\/0007252300290039"},{"key":"ref_32","unstructured":"Furino, S., Miao, Y., and Yin, J. (1996). Frames and Resolvable Designs: Uses, Constructions and Existence, CRC Press."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2004.828150","article-title":"Permutation Arrays for Powerline Communication and Mutually Orthogonal Latin Squares","volume":"50","author":"Colbourn","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_34","unstructured":"Kl\u00f8ve, T. (, January November). A combinatorial problem motivated by a data transmission application. Proceedings of the Norsk Informatikkonf (NIK), Bod\u00f8, Norway."},{"key":"ref_35","first-page":"5","article-title":"An application of permutation arrays to block ciphers","volume":"145","author":"Colbourn","year":"2000","journal-title":"Congr. Numer."},{"key":"ref_36","unstructured":"Ferreira, H.C., and Han Vinck, A.J. (2000, January 24\u201328). Interference cancellation with permutation trellis codes. Proceedings of the 52nd IEEE Vehicular Technology Conference (VTS) Fall 2000, Boston, MA, USA."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1023\/B:DESI.0000029212.52214.71","article-title":"Constructions for Permutation Codes in Powerline Communications","volume":"32","author":"Chu","year":"2004","journal-title":"Des. Codes Cryptogr."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1002\/jcd.21661","article-title":"Isometry invariant permutation codes and mutually orthogonal Latin squares","volume":"27","author":"Janiszczak","year":"2019","journal-title":"J. Comb. Des."},{"key":"ref_39","unstructured":"Abel, R.J.R., Janiszczak, I., and Staszewski, R. (2024). Improvements for lower bounds of mutually orthogonal Latin squares of sizes 54, 96 and 108. arXiv."},{"key":"ref_40","first-page":"123","article-title":"The number of mutually orthogonal Latin squares\u2014A table up to order 10,000","volume":"ZW","author":"Brouwer","year":"1979","journal-title":"Math. Cent. Rep."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Colbourn, C.J., and Dinitz, J.H. (1996). The CRC Handbook of Combinatorial Designs, CRC Press. [1st ed.].","DOI":"10.1201\/9781420049954"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Abel, R.J.R., Colbourn, C.J., and Dinitz, J. (2007). Mutually orthogonal latin squares (MOLS). Handbook of Combinatorial Designs, CRC Press.","DOI":"10.1201\/9781420010541"},{"key":"ref_43","unstructured":"Street, A.P., and Street, D.J. (1987). Combinatorics of Experimental Design, Oxford University Press."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Beth, T., Jungnickel, D., and Lenz, H. (1999). Design Theory, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511549533"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"221","DOI":"10.2307\/1967920","article-title":"Euler Squares","volume":"23","author":"MacNeish","year":"1922","journal-title":"Ann. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1090\/S0002-9947-1960-0111695-3","article-title":"On the Construction of sets of Mutually Orthogonal Latin Squares and the Falsity of Euler\u2019s Conjecture","volume":"95","author":"Bose","year":"1960","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"189","DOI":"10.4153\/CJM-1960-016-5","article-title":"Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler\u2019s Conjecture","volume":"12","author":"Bose","year":"1960","journal-title":"Can. J. Math."},{"key":"ref_48","first-page":"149","article-title":"Chapter 5\u2014Recursive Constructions of Mutually Orthogonal Latin Squares","volume":"Volume 46","author":"Keedwell","year":"1991","journal-title":"Latin Squares"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0378-3758(00)00276-7","article-title":"Mutually orthogonal Latin squares: A brief survey of constructions","volume":"95","author":"Colbourn","year":"2001","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0012-365X(74)90148-4","article-title":"Concerning the number of mutually orthogonal Latin squares","volume":"9","author":"Wilson","year":"1974","journal-title":"Discret. Math."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0012-365X(80)90053-9","article-title":"New Wilson-type constructions of mutually orthogonal Latin squares","volume":"32","author":"Wojtas","year":"1980","journal-title":"Discret. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0012-365X(81)90261-2","article-title":"A general construction for group-divisible designs","volume":"33","author":"Stinson","year":"1981","journal-title":"Discret. Math."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0012-365X(82)90149-2","article-title":"More mutually orthogonal latin squares","volume":"39","author":"Brouwer","year":"1982","journal-title":"Discret. Math."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1002\/jcd.20121","article-title":"Concerning Eight Mutually Orthogonal Latin Squares","volume":"15","author":"Abel","year":"2007","journal-title":"J. Comb. Des."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF01388383","article-title":"Thwarts in Transversal Designs","volume":"5","author":"Colbourn","year":"1995","journal-title":"Des. Codes Cryptogr."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/0012-365X(89)90329-4","article-title":"Some new Matrices-minus-diagonal and MOLS","volume":"76","author":"Wojtas","year":"1989","journal-title":"Discret. Math."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0378-3758(96)00012-2","article-title":"Some direct constructions for incomplete transversal designs","volume":"56","author":"Colbourn","year":"1996","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_58","first-page":"69","article-title":"Some V(12, t) vectors and designs from difference and quasi-difference matrices","volume":"40","author":"Abel","year":"2008","journal-title":"Australas. J. Comb."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1002\/jcd.21298","article-title":"Four mutually orthogonal Latin Squares of order 14","volume":"20","author":"Todorov","year":"2012","journal-title":"J. Comb. Des."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1002\/jcd.21384","article-title":"Existence of Five MOLS of Orders 18 and 60","volume":"23","author":"Abel","year":"2015","journal-title":"J. Comb. Des."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Wallis, W.D. (1996). Making the Mols Table. Computational and Constructive Design Theory, Springer.","DOI":"10.1007\/978-1-4757-2497-4"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1002\/jcd.10070","article-title":"Concerning seven and eight mutually orthogonal latin squares","volume":"12","author":"Abel","year":"2004","journal-title":"J. Comb. Des."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/0012-365X(77)90058-9","article-title":"On seven mutually orthogonal Latin squares","volume":"20","author":"Wojtas","year":"1977","journal-title":"Discret. Math."},{"key":"ref_64","unstructured":"Hall, M. (1986). Orthogonal Latin Squares. Combinatorial Theory, John Wiley & Sons, Ltd.. Chapter 13."},{"key":"ref_65","first-page":"141","article-title":"Four pairwise orthogonal Latin squares of order 15","volume":"6","author":"Schellenberg","year":"1978","journal-title":"Ars Comb."},{"key":"ref_66","first-page":"63","article-title":"Four mutually orthogonal Latin Squares of order 20","volume":"27","author":"Todorov","year":"1989","journal-title":"Ars Comb."},{"key":"ref_67","first-page":"54","article-title":"Five pairwise orthogonal Latin squares of order 21","volume":"1","author":"Nazarok","year":"1991","journal-title":"Issled. Oper. ASU"},{"key":"ref_68","unstructured":"Abel, R.J.R. (1995). On the Existence of Balanced Incomplete Block Designs and Transversal Designs. [Ph.D. Thesis, School of Mathematics, Faculty of Science]."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0378-3758(95)00073-9","article-title":"Three mutually orthogonal idempotent Latin squares of orders 22 and 26","volume":"51","author":"Abel","year":"1996","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_70","first-page":"147","article-title":"Four MOLS of order 26","volume":"17","author":"Colbourn","year":"1995","journal-title":"J. Comb. Math. Comb. Comput."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/0097-3165(93)90093-N","article-title":"Four MOLS of order 20, 30, 38 and 44","volume":"64","author":"Abel","year":"1993","journal-title":"J. Comb. Theory A"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1094","DOI":"10.1016\/j.ejc.2004.04.013","article-title":"Some difference matrix constructions and an almost completion for the existence of triplewhist tournaments TWh(v)","volume":"26","author":"Abel","year":"2005","journal-title":"Eur. J. Comb."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1002\/(SICI)1520-6610(1996)4:2<153::AID-JCD7>3.0.CO;2-E","article-title":"Five mutually orthogonal Latin squares of order 35","volume":"4","author":"Wojtas","year":"1996","journal-title":"J. Comb. Des."},{"key":"ref_74","first-page":"175","article-title":"Some new MOLS of order 2n\u00a0p for p a prime power","volume":"10","author":"Abel","year":"1994","journal-title":"Australas. J. Comb."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Wallis, W.D. (2003). The existence of 2-SOLSSOMs. Designs 2002: Further Computational and Constructive Design Theory, Kluwer.","DOI":"10.1007\/978-1-4613-0245-2"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1002\/(SICI)1520-6610(2000)8:3<218::AID-JCD7>3.0.CO;2-8","article-title":"Three new constructions of mutually orthogonal Latin squares","volume":"8","author":"Wojtas","year":"2000","journal-title":"J. Comb. Des."},{"key":"ref_77","first-page":"473","article-title":"Some mutually orthogonal Latin squares","volume":"19","author":"Mills","year":"1977","journal-title":"Congr. Numer."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/0097-3165(92)90052-V","article-title":"Sets of mutually orthogonal latin squares with \u201clike subsquares\u201d","volume":"61","author":"Roberts","year":"1992","journal-title":"J. Comb. Theory (Ser. A)"},{"key":"ref_79","first-page":"207","article-title":"Six MOLS of order 76","volume":"19","author":"Colbourn","year":"1995","journal-title":"J. Comb. Math. Comb. Comput."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0195-6698(80)80019-9","article-title":"A Series of Separable Designs with Application to Pairwise Orthogonal Latin Squares","volume":"1","author":"Brouwer","year":"1980","journal-title":"Eur. J. Comb."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1006\/ffta.1996.0018","article-title":"More Thwarts in Transversal Designs","volume":"2","author":"Colbourn","year":"1996","journal-title":"Finite Fields Their Appl."},{"key":"ref_82","unstructured":"The GAP Group (2024, November 07). GAP\u2014Groups, Algorithms, and Programming, Version 4.13.1. Available online: https:\/\/www.gap-system.org."},{"key":"ref_83","unstructured":"Brouwer, A.E. (2024, December 12). Recursive Constructions of Mutually orthogonal Latin Squares. Available online: https:\/\/www.win.tue.nl\/~aeb\/preprints\/6367A.pdf."},{"key":"ref_84","first-page":"485","article-title":"Baer Subplanes","volume":"47","author":"Salzmann","year":"2003","journal-title":"Ill. J. Math."},{"key":"ref_85","unstructured":"Hirschfeld, J.W.P. (1979). Projective Geometries over Finite Fields, Oxford University Press."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1090\/S0002-9947-1953-0054978-4","article-title":"Finite Projective Plane Geometries and Difference Sets","volume":"74","author":"Berman","year":"1953","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"549","DOI":"10.4153\/CMB-1964-051-8","article-title":"Finite projective planes with affine subplanes","volume":"7","author":"Ostrom","year":"1964","journal-title":"Can. Math. Bull."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. 1. The user language","volume":"24","author":"Cosma","year":"1997","journal-title":"J. Symb. Comput."},{"key":"ref_89","unstructured":"(2024, November 07). Magma Computational Algebra Systems V2.28-14. Available online: http:\/\/magma.maths.usyd.edu.au\/magma\/."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/S0195-6698(84)80028-1","article-title":"On the Geometry of Planar Difference Sets","volume":"5","author":"Jungnickel","year":"1984","journal-title":"Eur. J. Comb."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/S0021-9800(69)80095-5","article-title":"Some Maximal Arcs in Finite Projective Planes","volume":"6","author":"Denniston","year":"1969","journal-title":"J. Comb. Theory"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1002\/jcd.21393","article-title":"A Three-Factor Product Construction for Mutually Orthogonal Latin Squares","volume":"23","author":"Dukes","year":"2015","journal-title":"J. Comb. Des."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/j.disc.2014.11.018","article-title":"Some constructions for t pairwise orthogonal diagonal Latin squares based on difference matrices","volume":"338","author":"Abel","year":"2015","journal-title":"Discret. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1678\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:54:53Z","timestamp":1760115293000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1678"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,18]]},"references-count":93,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121678"],"URL":"https:\/\/doi.org\/10.3390\/sym16121678","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,18]]}}}