
-
Previous Article
Exponential generalised network descriptors
- AMC Home
- This Issue
-
Next Article
More cyclotomic constructions of optimal frequency-hopping sequences
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 $\mathcal{A}_q(5,3) = 2(q^3+1)$.
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. |
[1] |
Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese. Combining subspace codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021007 |
[2] |
Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. |
[3] |
Abdeslem Hafid Bentbib, Smahane El-Halouy, El Mostafa Sadek. Extended Krylov subspace methods for solving Sylvester and Stein tensor equations. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021026 |
[4] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
[5] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
[6] |
Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79 |
[7] |
Emily McMillon, Allison Beemer, Christine A. Kelley. Extremal absorbing sets in low-density parity-check codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021003 |
[8] |
Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021002 |
[9] |
Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 |
[10] |
Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020135 |
[11] |
Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077 |
[12] |
Muhammad Ajmal, Xiande Zhang. New optimal error-correcting codes for crosstalk avoidance in on-chip data buses. Advances in Mathematics of Communications, 2021, 15 (3) : 487-506. doi: 10.3934/amc.2020078 |
[13] |
Yun Gao, Shilin Yang, Fang-Wei Fu. Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes. Advances in Mathematics of Communications, 2021, 15 (3) : 387-396. doi: 10.3934/amc.2020072 |
[14] |
Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020133 |
[15] |
Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079 |
[16] |
Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075 |
2019 Impact Factor: 0.734
Tools
Metrics
Other articles
by authors
[Back to Top]