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LitNet Akademies Jaargang 9 (2), April 2012 [3] F.E. Bennet en L. Zhu. Further results on the existence of HSOLSSOM(h n ). Australas. J. Combin., 14:207–220, 1996. [4] F.E. Bennet en L. Zhu. The spectrum of HSOLSSOM(h n ) where h is even. Discrete Math., 158:11–25, 1996. [5] R.K. Brayton, D. Coppersmith, en A.J. Hoffman. Self-orthogonal Latin squares of all orders. Bull. Amer. Math. Soc., 80(1):116–118, 1974. [6] A.P. Burger, M.P. Kidd, en J.H. van Vuuren. A repository of self-orthogonal Latin squares. http://www.vuuren.co.za/ → Repositories. [7] A.P. Burger, M.P. Kidd, en J.H. Van Vuuren. Enumerasie van self-ortogonale latynse vierkante van orde 10. LitNet Akademies (Natuurwetenskappe), 7(3):1–22, 2010. [8] A.P. Burger, M.P. Kidd, en J.H. van Vuuren. Enumeration of isomorphism classes of self-orthogonal Latin squares. Ars Combin., 97:143–152, 2010. [9] A.P. Burger en J.H. van Vuuren. Skedulering van gade-vermydende gemengde-dubbels rondomtalie-tennistoernooie. Die Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, 28(1):35–63, 2009. [10] C.J. Colbourn en J.H. Dinitz. The CRC handbook of combinatorial designs. CRC Press, Boca Raton, 1996. [11] G.P. Graham en C.E. Roberts. Enumeration and isomorphic classification of selforthogonal Latin squares. J. Combin. Math. Combin. Comput., 59:101–118, 2006. [12] D.F. Holt. Handbook of computational group theory. CRC Press, Boca Raton, 2005. [13] P. Kaski en P.R.J. Österg˚ard. Classification algorithms for codes and designs. Springer, Berlin, 2006. [14] C.C. Lindner, R.C. Mullin, en D.R. Stinson. On the spectrum of resolvable orthogonal arrays invariant under the Klein group K 4. Aequationes Math., 26:176–183, 1983. [15] B.D. McKay, A. Meynert, en W. Myrvold. Small Latin squares, quasigroups and loops. J. Combin. Des., 15:98–119, 2007. [16] N.J.A. Sloane. The on-line encyclopedia of integer sequences. http://oeis.org. [17] W.D. Wallis. Combinatorial designs. Marcel Dekker, Inc, New York (NY), 1988. [18] S.P. Wang. On self-orthogonal Latin squares and partial transversals of Latin squares. PhD-proefskrif, Ohio State University, Columbus, 1978. [19] L. Zhu. A few more self-orthogonal Latin squares with symmetric orthogonal mates. Congr. Numer., 42:313–320, 1984. 20
Aanhangsel LitNet Akademies Jaargang 9 (2), April 2012 In hierdie aanhangsel verskaf ons volledige lyste van (ry, kolom)-paratoopklasverteenwoordigers van SOLVSOMs van ordes 4 n 9. Ons gee ook (1) die getal nie-(ry, kolom)paratope simmetriese Latynse vierkante wat ortogonaal is aan elke SOLV (ry, kolom)paratoopklasverteenwoordiger en (2) die getal nie-(ry, kolom)-paratope SOLVe ortogonaal aan elke (ry, kolom)-paratoopklasverteenwoordiger van simmetriese Latynse vierkante. Daar word in die onderstaande na SOLVe in dieselfde volgorde verwys as waarin hulle in die databasis in [6] verskyn. Vir n = 5 is daar slegs een SOLV (tot op (ry, kolom)-paratopisme na), en slegs een simmetriese maat is (tot op (ry, kolom)-paratopisme na) gevind wat ortogonaal is aan hierdie SOLV. Vir n = 7 is daar vier SOLVe, en slegs een simmetriese maat is vir die eerste en laaste van hierdie SOLVe gevind; geen simmetriese maat is vir die tweede en derde SOLVe gevind nie. Vir n = 8 is daar ook vier SOLVe; veertien simmetriese maats is (tot op (ry, kolom)-paratopisme na) vir die tweede van hierdie SOLVe gevind, terwyl agtien simmetriese maats (tot op (ry, kolom)-paratopisme na) vir die laaste van hierdie SOLVe gevind is (geen simmetriese maats is vir die eerste of derde SOLVe gevind nie). Vir n = 9 het ons slegs een simmetriese maat vir die k de SOLV gevind, waar k ∈ {38,68,73,86,88,92,101,102,133,135,137,139,140,141,144,155,158,163,165,169,171,173}, en twee nie-(ry, kolom)-paratope simmetriese maats vir die 172 ste en 175 ste SOLV in die databasis. Daar is slegs een unipotente, simmetriese Latynse vierkant van orde 4 tot op (ry, kolom)paratopisme na, en slegs een SOLV is (tot op (ry, kolom)-paratopisme na) gevind wat ortogonaal is aan hierdie simmetriese vierkant. Vir n = 8 is daar ses nie-(ry, kolom)-paratope, unipotente, simmetriese Latynse vierkante, wat ons in dieselfde volgorde oorweeg het as wat hulle (as een-faktorisasies) in Tabel VII.5.28 in [10, p. 743] voorkom. Vir elkeen van die eerste drie Latynse vierkante is daar twee nie-(ry, kolom)-paratope SOLVe gevind, terwyl slegs een (tot op (ry, kolom)-paratopisme na) vir die vyfde vierkant gevind is. Geen SOLVe is vir die vierde en sesde vierkante gevind nie. Die onderstaande lys bevat slegs (ry, kolom)-paratoopklasverteenwoordigers van idempotente, simmetriese Latynse vierkante van onewe orde wat ortogonale SOLV-maats toelaat. Onder elke vierkant toon ons in vetdruk die getal nie-(ry, kolom)-paratope SOLVe ortogonaal daaraan. n = 5 n = 7 n = 9 04123 0651243 021436587 021436587 021436587 021437856 021438765 021873456 051237846 081254367 41302 6104532 210573846 210758364 210758364 210684735 210754836 210467583 510478263 810726543 13240 5026314 102867453 102684735 102864735 102758364 102687354 102536847 102683457 102638754 20431 1463025 458301762 476301852 478301256 467301582 476301582 845301762 246301785 276301485 32014 2530461 376048215 358047216 356047812 385046217 358046217 763048215 378046512 523047816 1 4312650 637185024 684175023 684175023 748165023 847165023 376185024 783165024 468175032 3245106 584720631 537820641 537280641 873520641 783520641 458720631 824750631 357480621 2 845612370 863512470 863512470 536812470 635812470 584612370 465812370 645813270 763254108 745263108 745623108 654273108 564273108 637254108 637524108 734562108 11 2 1 1 5 3 2 1 Vervolgens lys ons die (ry, kolom)-paratoopklasverteenwoordigers van SOLVSOMs in standaardvorm. ’n SOLVSOM word as ’n paar Latynse vierkante langs mekaar gegee — die SOLV 21
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