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Proper 2-Covers of PG(3,q), q Even

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Abstract

A k-cover of Σ=PG(3q) is a set S of lines of Σ such that every point is on exactly k lines of S. S is proper if it contains no spread. The existence of proper k-covers of Σ is necessary for the existence of maximal partial packings of q 2+q+1−k spreads of Σ. Here we give the first construction of proper 2-packings of PG(3,q) with q even; for q odd these have been constructed by Ebert.

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References

  1. André, J.: Ñber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z. 60 (1954), 156–186.

    Google Scholar 

  2. Blokhuis, A. and Wilbrink, H.: manuscript 1994.

  3. Beutelspacher, A.: On parallelisms in finite projective spaces, Geom. Dedicata 3 (1974), 35–40.

    Google Scholar 

  4. Bruck, R. H.: Construction problems of finite projective planes, In: Conference on Combinatorial Mathematics and its Applications, University of North Carolina Press, 1970, pp. 426–514.

  5. Bruen, A. A. and Drudge, K.: The construction of Cameron–Liebler line classes in PG(3,q), q odd, Finite Fields Appl. 5(1) (1999), 35–45.

    Google Scholar 

  6. Cameron, P. J. and Liebler, R. A.: Tactical decompositions and orbits of projective groups, Linear Algebra Appl. 46 (1982), 91–102.

    Google Scholar 

  7. Ebert, G. L.: Partitioning projective geometries into caps, Canad. J. Math. 39(6) (1985), 1163–1175.

    Google Scholar 

  8. Ebert, G. L.: The completion problem for partial packings, Geom. Dedicata 18(2) (1985), 261–267.

    Google Scholar 

  9. Eisfeld, J.: On the Common Nature of Spreads and Pencils in PG(d,q), Discrete Math. 189 (1998), 95–104.

    Google Scholar 

  10. Glynn, D. G.: On a set of lines of PG(3,q) corresponding to a maximal cap contained in the Klein quadric of PG(5,q), Geom. Dedicata 26 (1988), 273–280.

    Google Scholar 

  11. Hirschfeld, J. W. P.: Finite Projective Spaces of Three Dimensions, Clarendon Press, Oxford (1985).

    Google Scholar 

  12. Thas, J.: Ovoidal translation planes, Arch. Math. 23 (1972), 110–112.

    Google Scholar 

  13. Tits, J.: Ovoïdes et groupes de Suzuki, Arch. Math. 13 (1962), 187–198.

    Google Scholar 

  14. Segre, B.: On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arith. 5, 315–332.

Download references

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Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road

    Keldon Drudge

  2. London, E1 4NS, U.K

    Keldon Drudge

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  1. Keldon Drudge
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Drudge, K. Proper 2-Covers of PG(3,q), q Even. Geometriae Dedicata 80, 59–64 (2000). https://doi.org/10.1023/A:1005223815861

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