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The Non-Existence of Finite Projective Planes of Order 10

Published online by Cambridge University Press:  20 November 2018

C. W. H. Lam
Affiliation:
Concordia University, Montréal, Québec
L. Thiel
Affiliation:
Concordia University, Montréal, Québec
S. Swiercz
Affiliation:
Concordia University, Montréal, Québec
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A finite projective plane of order n, with n > 0, is a collection of n 2+ n + 1 lines and n2 + n + 1 points such that

1. every line contains n + 1 points,

2. every point is on n + 1 lines,

3. any two distinct lines intersect at exactly one point, and

4. any two distinct points lie on exactly one line.

It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Assmus, E. F., Jr. and Mattson, H. F., Jr., On the possibility of a projective plane of order 10, Algebraic Theory of Codes II, Air Force Cambridge Research Laboratories Report AFCRL- 71–0013, Sylvania Electronic Systems, Needham Heights, Mass. (1970).CrossRefGoogle Scholar
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11. Lam, C., Crossfield, S.and Thiel, L., Estimates of a computer search for a projective plane of order 10, Congressus Numerantium 48 (1985), 253263.Google Scholar
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13. Lam, C. W. H. and Thiel, L. H., Backtrack search with isomorph rejection and consistency check, J. of Symbolic Computation, 7 (1989), 473485.Google Scholar
14. Lam, C. W. H., Thiel, L. H. and Swiercz, S., A computer search for a projective plane of order 10, in Algebraic, extremal and metric combinatorics 1986, London Mathematical Society, Lecture Notes Series 131 (1988), 155165.Google Scholar
15. MacWilliams, J., Sloane, N. J. A. and Thompson, J. G., On the existence of a projective plane of order 10, J. Comb. Theory, Series A., 14 (1973), 6678.Google Scholar
16. Mallows, C. L. and Sloane, N. J. A., Weight enumerators of self-orthogonal codes, Discrete Math. 9 (1974), 391400.Google Scholar
17. Thiel, L. H., Lam, C.and Swiercz, S., Using a CRAY-1 to perform backtrack search, Proc. of the Second International Conference on Supercomputing, San Francisco 3 (1987), 9299.Google Scholar
18. Tarry, G., Le problème des 36 officiers, C. R. Assoc. Fran. Av. Sci. 1 (1900), 122123, 2 (1901), 170203.Google Scholar
19. Whitesides, S. H., Collineations of projective planes of order 10, Parts I and II, J. Comb. Theory, Series A 26 (1979), 249277.Google Scholar
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