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Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

Published online by Cambridge University Press:  20 November 2018

R. C. Bose
Affiliation:
University of North Carolina, Case Institute of Technology and Remington Rand Univac, St. Paul, Minnesota
S. S. Shrikhande
Affiliation:
University of North Carolina, Case Institute of Technology and Remington Rand Univac, St. Paul, Minnesota
E. T. Parker
Affiliation:
University of North Carolina, Case Institute of Technology and Remington Rand Univac, St. Paul, Minnesota
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If

is the prime power decomposition of an integer v, and we define the arithmetic function n(v) by

then it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated as

MacNeish conjectured that the actual value of N(v) is n(v).

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Bose, R.C., On the construction of balanced incomplete block designs, Ann. Eugen. London, 9 (1939), 353399.CrossRefGoogle Scholar
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5. Bose, R.C., Shrikhande, S.S., and Bhattacharya, K., On the construction of group divisible incomplete block designs, Ann. Math. Stat., 24 (1953), 167195.CrossRefGoogle Scholar
6. Bose, R.C., Shrikhande, S.S., and Bhattacharya, K. On the construction of pairwise orthogonal Latin squares and the falsity of a conjecture of Ruler, U. N. C , Institute of Statistics, Mimeo. series no. 222. To be published in Trans. Amer. Math. Soc.Google Scholar
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9. Levi, F.W., Finite geometrical systems (University of Calcutta, 1942).Google Scholar
10. MacNeish, H.F., Ruler squares, Ann. Math., 28 (1922), 221227.CrossRefGoogle Scholar
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12. Parker, E.T., Construction of some sets of pairwise orthogonal Latin squares, Amer. Math. Soc. Notices 5 (1958), 815 (abstract). (To be published in Proc. Amer. Math. Soc. under the title: Construction of some sets of mutually orthogonal Latin squares.) Google Scholar
13. Parker, E.T. Orthogonal Latin squares, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 859862.Google ScholarPubMed
14. Rao, C.R., Factorial experiments derivable from combinatorial arrangements of arrays, J. Roy. Stat. Soc. Suppl., 9 (1947), 128139.CrossRefGoogle Scholar
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