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On the Spectrum of the Sizes of Maximal Partial Line Spreads in PG(2n,q), n ≥ 3

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A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Szőnyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N≥ 5, are given. In PG(2n,q), n ≥ 3, the largest possible size for a partial line spread is q2n-1+q2n-3+...+q3+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Szőnyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q2n+1−q)/(q2−1)−q3+q2−2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q ≥ 9.

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Authors and Affiliations

  1. J. Eisfeld, Taunushöhe 27, 65779, Kelkheim, Germany

    J. Eisfeld

  2. Department of Pure Maths and Computer Algebra, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    L. Storme

  3. Eötvös University, Budapest, Pázmány P. sétány 1/c, Budapest, Hungary, H-1117

    P. Sziklai

Authors
  1. J. Eisfeld
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  2. L. Storme
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  3. P. Sziklai
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Correspondence to L. Storme.

Additional information

J. Eisfeld: Supported by the FWO Research Network WO.011.96N

P. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.

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Eisfeld, J., Storme, L. & Sziklai, P. On the Spectrum of the Sizes of Maximal Partial Line Spreads in PG(2n,q), n ≥ 3. Des Codes Crypt 36, 101–110 (2005). https://doi.org/10.1007/s10623-004-1165-y

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  • DOI: https://doi.org/10.1007/s10623-004-1165-y

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