Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T00:39:04.618Z Has data issue: false hasContentIssue false
  • English
  • Français

The Non-Existence of Finite Projective Planes of Order 10

Published online by Cambridge University Press:  20 November 2018

C. W. H. Lam ,
L. Thiel  and
S. Swiercz
C. W. H. Lam
Affiliation:
Concordia University, Montréal, Québec
L. Thiel
Affiliation:
Concordia University, Montréal, Québec
S. Swiercz
Affiliation:
Concordia University, Montréal, Québec
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A finite projective plane of order n, with n > 0, is a collection of n2+ 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
Information
Canadian Journal of Mathematics , Volume 41 , Issue 6 , 01 December 1989 , pp. 1117 - 1123
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).Google Scholar
2. Anstee, R. P., Hall, M., Jr. and Thompson, J. G., Planes of order 10 do not have a collineation of order 5, J. Comb. Theory, Series A, 29 (1980), 39–58.Google Scholar
3. Bose, R. C., On the application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares, Sankhyã 3 (1938), 323–338.Google Scholar
4. Brack, R. H. and Ryser, H. J., The non-existence of certain finite projective planes, Can. J. Math. 1 (1949), 88–93.Google Scholar
5. Bruen, A.and Fisher, J. C., Blocking Sets, k-arcs and nets of order ten, Advances in Math. 10 (1973), 317–320.Google Scholar
6. Carter, J. L., On the existence of a projective plane of order ten, Ph.D. thesis, Univ. of Calif., Berkeley (1974).Google Scholar
7. Denniston, R. H. F., Non-existence of a certain projective plane, J. Austral. Math. Soc. 10 (1969), 214–218.Google Scholar
8. Hall, M., Jr., Configurations in a plane of order 10, Ann. Discrete Math. 6 (1980), 157–174.Google Scholar
9. Janko, Z.and van Trung, T., Projective planes of order 10 do not have a collineation of order 3, J. Reine Angew. Math. 325 (1981), 189–209.Google Scholar
10. Lam, C. W. H., Thiel, L., Swiercz, S.and McKay, J., The nonexistence of ovals in a projective plane of order 10, Discrete Mathematics 45 (1983), 319–321.Google Scholar
11. Lam, C., Crossfield, S.and Thiel, L., Estimates of a computer search for a projective plane of order 10, Congressus Numerantium 48 (1985), 253–263.Google Scholar
12. Lam, C. W. H., Thiel, L.and Swiercz, S., The nonexistence of code words of weight 16 in a projective plane of order 10, J. Comb. Theory, Series A, 42 (1986), 207–214.Google Scholar
13. Lam, C. W. H. and Thiel, L. H., Backtrack search with isomorph rejection and consistency check, J. of Symbolic Computation, 7 (1989), 473–485.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), 155–165.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), 66–78.Google Scholar
16. Mallows, C. L. and Sloane, N. J. A., Weight enumerators of self-orthogonal codes, Discrete Math. 9 (1974), 391–400.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), 92–99.Google Scholar
18. Tarry, G., Le problème des 36 officiers, C. R. Assoc. Fran. Av. Sci. 1 (1900), 122–123, 2 (1901), 170–203.Google Scholar
19. Whitesides, S. H., Collineations of projective planes of order 10, Parts I and II, J. Comb. Theory, Series A 26 (1979), 249–277.Google Scholar