Figure 1.
Construction for $ q $ odd
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August
2019, 13(3): 393-404.
doi: 10.3934/amc.2019025
Optimal subspace codes in $ {{\rm{PG}}}(4,q) $
| 1. | Department of Mathematics, Informatics and Economics, University of Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy |
| 2. | Dipartimento di Mechanics, Mathematics and Management, Polytechnic University of Bari, Via Orabona 4, 70125 Bari, Italy |
| 3. | Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281 - Building S8, 9000 Ghent, Belgium |
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 $\mathcal{A}_q(5,3) = 2(q^3+1)$.
Keywords:
Galois geometry,
subspace codes.
Mathematics Subject Classification:
Primary: 05B25; Secondary: 51E20, 94B25.
Citation:
Antonio Cossidente, Francesco Pavese, Leo Storme. Optimal subspace codes in $ {{\rm{PG}}}(4,q) $. Advances in Mathematics of Communications,
2019, 13
(3)
: 393-404.
doi: 10.3934/amc.2019025
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References:
| [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. Google Scholar |
| [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. Honold, M. 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. Honold, M. 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. |
show all references
References:
| [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. Google Scholar |
| [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. Honold, M. 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. Honold, M. 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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