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On the convex hull of projective planes

Published online by Cambridge University Press:  20 August 2008

Jean-François Maurras
Affiliation:
Laboratoire d'Informatique Fondamentale de Marseille, France, [email protected]
Roumen Nedev
Affiliation:
Technical University - Sofia, FKSU, bld. K. Ohridski 8, Sofia 1000, Bulgaria; [email protected]
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Abstract

We study the finite projective planes with linear programmingmodels. We give a complete description of the convex hull of thefinite projective planes of order 2. We give some integer linearprogramming models whose solution are, either a finiteprojective (or affine) plane of order n, or a (n+2)-arc.

Type
Research Article
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

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References

Anglada, O. and Maurras, J.F., Enveloppe convexe des hyperplans d'un espace affine fini, avec Olivier Anglada. RAIRO-Oper. Res. 37 (2003) 213219. CrossRef
D. Avis, http://cgm.cs.mcgill.ca/ avis/C/lrs.html.
Avis, D. and Fukuda, K., A pivoting algorithm for convex hulls and vertex enumeration of arrankements and polyhedra. Discrete Comput. Geom. 8 (1992) 295313. CrossRef
I. Bárány and Pór, 0-1 polytopes with many facets, Adv. Math. 161 (2001) 209–228.
Bruck, R.H. and Ryser, H.J., The nonexistence of certain finite projective planes. Can. J. Math. 1 (1949) 8893. CrossRef
F.C. Bussemaker and J.J. Seidel, Symmetric Hadamard matrices of order 36. Report 70-WSK-02, TH Eindhoven, July (1970).
T. Christof, www.zib.de/Optimization/Software/porta.
K. Fukuda, http://cs.mcgill.ca/ fukuda/soft/cdd.
P.B. Gibbons, Computing Techniques for the Construction and Analysis of Block Designs, Techn. Report N°92, Dept. of Computer Science, University of Toronto (1976).
Kirkman, T.R., On a problem in combinations. Camb. Dublin Math. J. 2 (1847) 191204.
Lam, C.W.H., The Search for a Finite Projective Plane of Order 10. Am. Math. Mon. 98 (1991) 305318. CrossRef
Limbos, M., Projective embeddings of small Steiner triple systems. Ann. Discrete Math. 7 (1980) 151173. CrossRef
Mathon, R.A, Phelps, K.T. and Rosa, A., Small Steiner triple systems and their properties. Ars Combinatoria 15 (1983) 3110.
Maurras, J.F., An exemple of dual polytopes in the unit hypercube. Ann. Discrete Math. 1 (1977) 391392. CrossRef
Maurras, J.F., The Line Polytope of a finite Affine Plane. Discrete Math. 115 (1993) 283286. CrossRef
T.S. Motzkin, H. Raiffa, G.L. Thompson and R.M. Thrall, The double description method, in H.W. Kuhn and A.W. Tucker, Eds., Contributions to theory of games, Vol. 2, Princeton University Press, Princeton (1953).
H.S. White, F.N. Cole and L.D. Cummings, Complete classification of the triad systems on fifteen elements. Mem. Nat. Acad. Sci. U.S.A. 14, 2nd memoir (1919) 1–89.