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Incomplete self-orthogonal latin squares

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

Katherine Heinrich  and
L. Zhu
Katherine Heinrich
Affiliation:
Department of MathematicsSimon Fraser UniversityBurnaby, B. C., V5A 1SB, Canada
L. Zhu
Affiliation:
Department of MathematicsSuzhou UniversitySuzhou People's Republic of, China
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Abstract

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We show that for all n ≥ 3k + 1, n ≠ 6, there exists an incomplete self-orthogonal latin square of order n with an empty order k subarray, called an ISOLS(n;k), except perhaps when (n;k) ∈ {(6m + i;2m):i = 2, 6}.

MSC classification

Secondary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 365 - 384
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bennett, F. E., Self-orthogonal latin squares with self-orthogonal subsquares, unpublished manuscript.Google Scholar
[2]Bennett, F. E. and Mendelsohn, N. S., ‘On the spectrum of Stein quasigroups’, Bull. Austral. Math. Soc. 21 (1980), 47–63.CrossRefGoogle Scholar
[3]Brayton, R. B., Coppersmith, D. and Hoffman, A. J., Self-orthogonal latin squares of all orders n ≠ 2, 3, 6 (Coll. Internationale sulle Teorie Combinatorie, Roma, 1973, Atti dei convegni Lincei, No. 17, Tomo II, 1976), pp. 509–517.Google Scholar
[4]Brouwer, A. E. and van Rees, G. H. J., ‘More mutually orthogonal latin squares’, Discrete Math. 39 (1982), 263–281.CrossRefGoogle Scholar
[5]Crampin, D. J. and Hilton, A. J. W., ‘On the spectra of certain types of latin square’, J. Combinatorial Theory 19 (1975), 84–94.CrossRefGoogle Scholar
[6]Dénes, J. and Keedwell, A. D., Latin squares and their applications (Akadémiai Kiadó, Budapest, 1974).Google Scholar
[7]Dinitz, J. H. and Stinson, D. R., ‘Mols with holes’, Discrete Math. 44 (1983), 145–154.CrossRefGoogle Scholar
[8]Drake, D. A. and Larson, J. A., ‘Pairwise balanced designs whose line sizes do not divide six’, J. Combin. Theory Ser. A 34 (1983), 266–300.CrossRefGoogle Scholar
[9]Drake, D. A. and Lenz, H., ‘Orthogonal latin squares with orthogonal subsquares’, Archiv der Mathematik 34 (1980), 565–576.CrossRefGoogle Scholar
[10]Hedayat, A., ‘A generalization of sum composition: self-orthogonal latin square design with sub self orthogonal latin square designs’, J. Combin. Theory Ser. A 24 (1978), 202–210.CrossRefGoogle Scholar
[11]Hedayat, A. and Seiden, E., ‘On the theory and application of sum composition of latin squares and orthogonal latin squares’, Pacific J. Math. 54 (1974), 85–113.CrossRefGoogle Scholar
[12]Heinrich, K., ‘Self-orthogonal latin squares with self-orthogonal subsquares’, Ars Combinatoria 3 (1977), 251–266.Google Scholar
[13]Heinrich, K., Latin squares with and without subsquares of prescribed type, Ann. Discrete Math., to appear.Google Scholar
[14]Heinrich, K. and Zhu, L., ‘Existence of orthogonal latin squares with aligned subsquares’, Discrete Math. 59 (1986), 69–78.CrossRefGoogle Scholar
[15]Horton, J. D., ‘Sub-latin squares and incomplete orthogonal arrays’, J. Combin. Theory 16 (1974), 23–33.CrossRefGoogle Scholar
[16]Lindner, C. C., Mullin, R. C. and Stinson, D. R., ‘On the spectrum of resolvable orthogonal arrays invariant under the Klein group K4’, Aequationes Math. 26 (1983), 176–183.CrossRefGoogle Scholar
[17]Parker, E. T., ‘Orthogonal latin squares’, Proc. Nat. Acad. Sci. U. S. A. 45 (1959), 859–862.CrossRefGoogle ScholarPubMed
[18]Wallis, W. D. and Zhu, L., Orthogonal latin squares with small subsquares (Combinatorial Mathematics X (Adelaide 1982), Lecture Notes in Math., Vol. 1036, Springer-Verlag, Berlin and New York, 1983), pp. 398–409.Google Scholar
[19]Wang, S. M. P., On self-orthogonal latin squares and partial transversals of latin squares (Ph. D. thesis, Ohio State University, Columbus, Ohio, 1978).Google Scholar
[20]Zhu, L., ‘A short disproof of Euler's conjecture concerning orthogonal latin squares’, Ars Combinatoria 14 (1982), 47–55.Google Scholar
[21]Zhu, L., ‘Some results on orthogonal latin squares with orthogonal subsquares’, Utilitas Math. 25 (1984), 241–248.Google Scholar
[22]Zhu, L., ‘Orthogonal latin squares with subsquares’, Discrete Math. 48 (1984), 315–321.CrossRefGoogle Scholar
[23]Zhu, L., ‘A few more self-orthogonal latin squares with symmetric orthogonal mates’, Congr. Numer. 42 (1984), 313–320.Google Scholar