Existence of Projective Planes

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Perrott, Xander (2016)
  • Subject: Mathematics - Combinatorics | Mathematics - History and Overview

This report gives an overview of the history of finite projective planes and their properties before going on to outline the proof that no projective plane of order 10 exists. The report also investigates the search carried out by MacWilliams, Sloane and Thompson in 1970 [12] and confirms their result by providing independent verification that there is no vector of weight 15 in the code generated by the projective plane of order 10.
  • References (14)
    14 references, page 1 of 2

    [1] E. F. Assmus, Jr. and H. F. Mattson, 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, Needleham Heights, Mass., 1970.

    [2] R. C. Bose, On the application of the properties of Galois elds to the problem of construction of hyper-Graeco-Latin squares, Sankhya, 3 (1938) 323-338.

    [3] R. H. Bruck and H. J. Ryser, The non-existence of certain projective planes, Can. J. Math., 1 (1949) 88-93.

    [4] A. Bruen and J. C. Fisher, Blocking sets, k-arcs and Nets of Order Ten, Advances in Math., 10 (1973) 317-320.

    [5] R. H. F. Denniston, Non-existence of a Certain Projective Plane, J. Austral. Math. Soc., 10 (1969) 214-218.

    [6] L. Euler, Recherches sur une nouvelle espece de quarres magiques, Verh. Zeeuwsch. Genootsch. Wetensch Vlissengen, 9 (1782) 85-239.

    [7] J. Karhstrom, On Projective Planes, Bachelor's Thesis, Mid Sweden University, 2002.

    [8] C. W. H. Lam, The Search for a Finite Projective Plane of Order 10, Amer. Math. Monthly, 98 (1991) 305-318.

    [9] C. W. H. Lam, L. Thiel, and S. Swiercz, The non-existence of nite projective planes of order 10, Can. J. Math.,XLI (1989) 1117-1123.

    [10] C. W. H. Lam, L. Thiel, and S. Swiercz, The Nonexistence of Code Words of Weight 16 in a Projective Plane of Order 10, J. Comb. Theory, Series A., 42 (1986) 207-214.

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