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Optimal subspace codes in $ {{\rm{PG}}}(4,q) $

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  • We investigate subspace codes whose codewords are subspaces of ${\rm{PG}}(4,q)$ having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that $\mathcal{A}_q(5,3) = 2(q^3+1)$.

    Mathematics Subject Classification: Primary: 05B25; Secondary: 51E20, 94B25.

    Citation:

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  • Figure 1.  Construction for $ q $ odd

    Figure 2.  Construction for $ q $ even

    Figure 3.  Construction in PG$ (5,q) $, $ q $ even

    Figure 4.  Construction in the hyperplane $ S $ of PG$ (5,q) $, $ q $ even

  • [1] A. Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math. Z., 145 (1975), 211-229.  doi: 10.1007/BF01215286.
    [2] J. D'haeseleer, Subspace Codes en Hun Meetkundige Achtergrond, Master project Ghent University, Academic year 2016-2017.
    [3] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.
    [4] J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1985.
    [5] T. HonoldM. Kiermaier and S. Kurz, Constructions and bounds for mixed–dimension subspace codes, Adv. Math. Commun., 10 (2016), 649-682.  doi: 10.3934/amc.2016033.
    [6] T. HonoldM. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, Contemp. Math., 632 (2015), 157-176.  doi: 10.1090/conm/632/12627.
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