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Repeated-root constacyclic codes of length $ 6lp^s $
Properties of sets of Subspaces with Constant Intersection Dimension
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium |
A $ (k,k-t) $-SCID (set of Subspaces with Constant Intersection Dimension) is a set of $ k $-dimensional vector spaces that have pairwise intersections of dimension $ k-t $. Let $ \mathcal{C} = \{\pi_1,\ldots,\pi_n\} $ be a $ (k,k-t) $-SCID. Define $ S: = \langle \pi_1, \ldots, \pi_n \rangle $ and $ I: = \langle \pi_i \cap \pi_j \mid 1 \leq i < j \leq n \rangle $. We establish several upper bounds for $ \dim S + \dim I $ in different situations. We give a spectrum result under certain conditions for $ n $, giving examples of $ (k,k-t) $-SCIDs reaching a large interval of values for $ \dim S + \dim I $.
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S.-Y. R. Li, R. W. Yeung and N. Cai,
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[13] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
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K. Metsch and L. Storme,
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show all references
References:
[1] |
R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung,
Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.
doi: 10.1109/18.850663. |
[2] |
R. D. Barrolleta, L. Storme, E. Suárez-Canedo and P. Vandendriessche,
On primitive constant dimension codes and a geometrical sunflower bound, Adv. Math. Commun., 11 (2017), 757-765.
doi: 10.3934/amc.2017055. |
[3] |
A. Beutelspacher,
Partial spreads in finite projective spaces and partial designs, Math. Zeit., 145 (1975), 211-229.
doi: 10.1007/BF01215286. |
[4] |
A. Beutelspacher and J. Ueberberg,
A characteristic property of geometric $t$-spreads in finite projective spaces, Europ. J. Combin., 12 (1991), 277-281.
doi: 10.1016/S0195-6698(13)80110-2. |
[5] |
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. |
[6] |
T. Etzion, Problems on $q$-analogs in coding theory, Preprint, arXiv: 1305.6126. Google Scholar |
[7] |
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. |
[8] |
E. Gorla and A. Ravagnani, Equidistant subspace codes, Linear Algebra Appl., 490 (2016), 48-65. Google Scholar |
[9] |
T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi and B. Leong,
A random linear network coding approach to multicast, IEEE Trans. Inform. Theory, 52 (2006), 4413-4430.
doi: 10.1109/TIT.2006.881746. |
[10] |
R. Kötter 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. |
[11] |
M. Lavrauw and G. Van de Voorde,
Field reduction and linear sets in finite geometry, Topics in finite fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 632 (2015), 271-293.
doi: 10.1090/conm/632/12633. |
[12] |
S.-Y. R. Li, R. W. Yeung and N. Cai,
Linear network coding, IEEE Trans. Inform. Theory, 49 (2003), 371-381.
doi: 10.1109/TIT.2002.807285. |
[13] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[14] |
K. Metsch and L. Storme,
Partial $t$-spreads in PG$(2t+1, q)$, Des. Codes Cryptogr., 18 (1999), 199-216.
doi: 10.1023/A:1008305824113. |
[15] |
O. Polverino,
Linear sets in finite projective spaces, Discrete Math., 310 (2010), 3096-3107.
doi: 10.1016/j.disc.2009.04.007. |
[16] |
B. Segre,
Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.
doi: 10.1007/BF02410047. |
[17] |
C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379–423,623–656.
doi: 10.1002/j.1538-7305.1948.tb01338.x. |
[18] |
A. Weil, Adeles and Algebraic Groups, Progress in Mathematics, 23. Birkhäuser, Boston, Mass., 1982. |
Condition | Upper bound |
Sharp? | Theorem |
yes | Theorem 2.1 & 2.2 | ||
unknown | Theorem 2.5 | ||
no | Theorem 2.1 & 2.2 |
Condition | Upper bound |
Sharp? | Theorem |
yes | Theorem 2.1 & 2.2 | ||
unknown | Theorem 2.5 | ||
no | Theorem 2.1 & 2.2 |
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