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

  • R. C. Bose (a1), S. S. Shrikhande (a1) and E. T. Parker (a1)
Extract

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).

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References
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1. Bose, R.C., On the construction of balanced incomplete block designs, Ann. Eugen. London, 9 (1939), 353399.
2. Bose, R.C., A note on the resolvability of balanced incomplete block designs, Sankhyâ, 6 (1942), 105110.
3. Bose, R.C. and Connor, W.S., Combinatorial properties of group divisible incomplete block designs, Ann. Math. Stat., 23 (1952), 367383.
4. Bose, R.C. and Shrikhande, S.S., On the falsity of Ruler's conjecture about the non-existence of two orthogonal Latin squares of order 4t + 2, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 734737.
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.
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.
7. Bush, K.A., Orthogonal arrays of index unity, Ann. Math. Stat., 28 (1952), 426434.
8. Euler, L., Recherches sur une nouvelle espèce des quarres magiques, Verh. zeeuwsch Genoot. Weten. Vliss., 9 (1782), 85239.
9. Levi, F.W., Finite geometrical systems (University of Calcutta, 1942).
10. MacNeish, H.F., Ruler squares, Ann. Math., 28 (1922), 221227.
11. Mann, H.B., The construction of orthogonal Latin squares, Ann. Math. Stat., 13 (1942), 418423.
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.)
13. Parker, E.T. Orthogonal Latin squares, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 859862.
14. Rao, C.R., Factorial experiments derivable from combinatorial arrangements of arrays, J. Roy. Stat. Soc. Suppl., 9 (1947), 128139.
15. Yates, F., Incomplete randomised blocks, Ann. Eugen. London, 7 (1936), 121140.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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