A completion problem for finite affine planes


A partial affine plane (PAP) of ordern is ann 2-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn 2+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn 2+n−√n can be completed and that forn≠3 any PAP withn 2+n−2 lines can be completed.

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  1. [1]

    M. R. Best, A. E. Brouwer, F. J. MacWilliams, A. M. Odlyzko andN. J. A. Sloane, Bounds for binary codes of length less than 25,IEEE Trans. Info. Theory 24 (1978), 81–93.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    R. H. Bruck, Finite nets II. Uniqueness and imbedding,Pacific J. Math. 13 (1963), 421–457.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    S. Dow, An improved bound for extending partial projective planes,Discrete Math. 45 (1983), 199–207.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    S. A. Vanstone, The extendibility of (r, 1)-designs,Proc. Third Manitoba Conf. on Numerical Math., Winnipeg (1973), 409–418.

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Dow, S. A completion problem for finite affine planes. Combinatorica 6, 321–325 (1986). https://doi.org/10.1007/BF02579258

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AMS subject classification (1980)

  • 05 B 25
  • 51 E 15