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The number of designs with geometric parameters grows exponentially

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Abstract

It is well-known that the number of 2-designs with the parameters of a classical point-hyperplane design PG n-1(n, q) grows exponentially. Here we extend this result to the number of 2-designs with the parameters of PG d (n, q), where 2 ≤ d ≤ n − 1. We also establish a characterization of the classical geometric designs in terms of hyperplanes and, in the special case d = 2, also in terms of lines. Finally, we shall discuss some interesting configurations of hyperplanes arising in designs with geometric parameters.

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Author information

Authors and Affiliations

  1. Lehrstuhl für Diskrete Mathematik, Optimierung und Operations Research, Universität Augsburg, 86135, Augsburg, Germany

    Dieter Jungnickel

  2. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Vladimir D. Tonchev

Authors
  1. Dieter Jungnickel
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  2. Vladimir D. Tonchev
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Corresponding author

Correspondence to Dieter Jungnickel.

Additional information

Communicated by Ron Mullin.

To Spyros Magliveras on the occasion of his 70th birthday.

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Jungnickel, D., Tonchev, V.D. The number of designs with geometric parameters grows exponentially. Des. Codes Cryptogr. 55, 131–140 (2010). https://doi.org/10.1007/s10623-009-9299-6

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  • DOI: https://doi.org/10.1007/s10623-009-9299-6

Keywords

Mathematics Subject Classifications (2000)