Figure 1.
The $(k,k-t)$ -SCID described in Example 1
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On the classification of $\mathbb{Z}_4$-codes
November
2017, 11(4): 757-765.
doi: 10.3934/amc.2017055
On primitive constant dimension codes and a geometrical sunflower bound
| 1. | Departament d'Enginyeria de la Informacio i de les Comunicacions, Edifici Q, Universitat Autónoma de Barcelona, 08193 -Bellaterra -Cerdanyola del Vallés (Barcelona), Spain |
| 2. | Department of Mathematics (WE01), Krijgslaan 281 -building S25, 9000 Gent, Belgium |
In this paper we study subspace codes with constant intersection dimension (SCIDs). We investigate the largest possible dimension spanned by such a code that can yield non-sunflower codes, and classify the examples attaining equality in that bound as one of two infinite families. We also construct a new infinite family of primitive SCIDs.
Keywords:
Subspace codes,
constant intersection dimension codes,
rank codes,
finite geometry,
Galois geometry.
Mathematics Subject Classification:
05B25, 51E20, 94B60.
Citation:
Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications,
2017, 11
(4)
: 757-765.
doi: 10.3934/amc.2017055
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References:
| [1] |
D. Bartoli and F. Pavese,
A note on equidistant subspace codes, Discrete Appl. Math., 198 (2016), 291-296.
doi: 10.1016/j.dam.2015.06.017. |
| [2] |
A. Beutelspacher, J. Eisfeld and J. Müller,
On sets of planes in ${\text{PG}}(d,q)$ intersecting mutually in one point, Geom. Dedicata, 78 (1999), 143-159.
doi: 10.1023/A:1005294416997. |
| [3] |
J. Eisfeld,
On sets of $n$-dimensional subspaces of projective spaces intersecting mutually in an $(n-2)$-dimensional subspace, Discrete Math., 255 (2002), 81-85.
doi: 10.1016/S0012-365X(01)00390-9. |
| [4] |
T. Etzion and N. Raviv,
Equidistant codes in the Grassmannian, Discrete Appl. Math., 186 (2015), 87-97.
doi: 10.1016/j.dam.2015.01.024. |
| [5] |
R. Koetter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
| [6] |
K. Metsch and L. Storme,
Partial $t$-spreads in ${\text{PG}}(2t+1,q)$, Des. Codes Cryptogr., 18 (1999), 199-216.
doi: 10.1023/A:1008305824113. |
| [7] |
B. Segre,
Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.
doi: 10.1007/BF02410047. |
show all references
References:
| [1] |
D. Bartoli and F. Pavese,
A note on equidistant subspace codes, Discrete Appl. Math., 198 (2016), 291-296.
doi: 10.1016/j.dam.2015.06.017. |
| [2] |
A. Beutelspacher, J. Eisfeld and J. Müller,
On sets of planes in ${\text{PG}}(d,q)$ intersecting mutually in one point, Geom. Dedicata, 78 (1999), 143-159.
doi: 10.1023/A:1005294416997. |
| [3] |
J. Eisfeld,
On sets of $n$-dimensional subspaces of projective spaces intersecting mutually in an $(n-2)$-dimensional subspace, Discrete Math., 255 (2002), 81-85.
doi: 10.1016/S0012-365X(01)00390-9. |
| [4] |
T. Etzion and N. Raviv,
Equidistant codes in the Grassmannian, Discrete Appl. Math., 186 (2015), 87-97.
doi: 10.1016/j.dam.2015.01.024. |
| [5] |
R. Koetter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
| [6] |
K. Metsch and L. Storme,
Partial $t$-spreads in ${\text{PG}}(2t+1,q)$, Des. Codes Cryptogr., 18 (1999), 199-216.
doi: 10.1023/A:1008305824113. |
| [7] |
B. Segre,
Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.
doi: 10.1007/BF02410047. |
Figure 2.
The $(k,k-t)$ -SCID described in Example 2
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