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

* Corresponding author

Received  July 2017 Revised  June 2018 Published  April 2019

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 $\mathcal{A}_q(5,3) = 2(q^3+1)$.

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
References:
[1]

A. Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math. Z., 145 (1975), 211-229.  doi: 10.1007/BF01215286.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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