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On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$
Constant dimension codes from Riemann-Roch spaces
1. | Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy |
2. | Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (Trento), Italy |
Some families of constant dimension codes arising from Riemann-Roch spaces associated with particular divisors of a curve $\mathcal{X}$ are constructed. These families are generalizations of the one constructed by Hansen.
References:
[1] |
D. Bartoli and F. Pavese,
A note on Equidistant subspaces codes, Discrete Appl. Math., 198 (2016), 291-296.
doi: 10.1016/j.dam.2015.06.017. |
[2] |
A. Cossidente and F. Pavese,
On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.
doi: 10.1007/s10623-014-0018-6. |
[3] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[4] |
T. Etzion and N. Silberstein,
Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.
doi: 10.1109/TIT.2012.2220119. |
[5] |
T. Etzion and A. Vardy,
Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[6] |
M. Gadouleau and Z. Yan,
Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.
doi: 10.1109/TIT.2010.2048447. |
[7] |
R. L. Graham, D. E. Knuth and O. Patashnik,
Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994. |
[8] |
J. P. Hansen,
Riemann-Roch theorem and linear network codes, Int. J. Math. Comput. Sci., 10 (2015), 1-11.
|
[9] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, in Topics in Finite Fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott Editors, Contemporary Mathematics, 632 (2015), 157-176. |
[10] |
A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, Cryptography and coding, in Lecture Notes in Compututer Science, Springer, Berlin, 5921 (2009), 1-21. |
[11] |
R. Koetter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. on Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[12] |
D. Silva, F. R. Kschischang and R. Koetter,
A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[13] |
A. L. Trautmann and J. Rosenthal, New improvements on the echelon-ferrers construction, in Proc. Int. Symp. on Math. Theory of Networks and Systems, 2010,405-408 Google Scholar |
show all references
References:
[1] |
D. Bartoli and F. Pavese,
A note on Equidistant subspaces codes, Discrete Appl. Math., 198 (2016), 291-296.
doi: 10.1016/j.dam.2015.06.017. |
[2] |
A. Cossidente and F. Pavese,
On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.
doi: 10.1007/s10623-014-0018-6. |
[3] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[4] |
T. Etzion and N. Silberstein,
Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.
doi: 10.1109/TIT.2012.2220119. |
[5] |
T. Etzion and A. Vardy,
Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[6] |
M. Gadouleau and Z. Yan,
Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.
doi: 10.1109/TIT.2010.2048447. |
[7] |
R. L. Graham, D. E. Knuth and O. Patashnik,
Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994. |
[8] |
J. P. Hansen,
Riemann-Roch theorem and linear network codes, Int. J. Math. Comput. Sci., 10 (2015), 1-11.
|
[9] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, in Topics in Finite Fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott Editors, Contemporary Mathematics, 632 (2015), 157-176. |
[10] |
A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, Cryptography and coding, in Lecture Notes in Compututer Science, Springer, Berlin, 5921 (2009), 1-21. |
[11] |
R. Koetter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. on Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[12] |
D. Silva, F. R. Kschischang and R. Koetter,
A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[13] |
A. L. Trautmann and J. Rosenthal, New improvements on the echelon-ferrers construction, in Proc. Int. Symp. on Math. Theory of Networks and Systems, 2010,405-408 Google Scholar |
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