Cyclic orbit codes and stabilizer subfields

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May 2015, 9(2): 177-197. doi: 10.3934/amc.2015.9.177

Cyclic orbit codes and stabilizer subfields

Heide Gluesing-Luerssen ^1, , Katherine Morrison ^2, and Carolyn Troha ^3,

1.	University of Kentucky, Department of Mathematics, Lexington, KY 40506-0027
2.	School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639, United States
3.	Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, United States

Received March 2014 Published May 2015

Abstract
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Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to the stabilizer of the generating subspace. Finally, with a linkage construction larger, and longer, constant dimension codes can be derived from cyclic orbit codes without compromising the distance.

Keywords: Random network coding, group actions., cyclic orbit codes, constant dimension subspace codes.

Mathematics Subject Classification: Primary: 11T71, 94B60; Secondary: 51E2.

Citation: Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177

References:

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[10]	A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in Proc. 12th IMA Conf. Crypt. Coding, Cirencester, 2009, 1-21. doi: 10.1007/978-3-642-10868-6_1.
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[12]	A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science (eds. J. Calmet, W. Geiselmann and J. Müller-Quade), Springer, Berlin, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4.
[13]	J. Rosenthal and A.-L. Trautmann, A complete characterization of irreducible cyclic orbit codes and their Plücker embedding, Des. Codes Crypt., 66 (2013), 275-289. doi: 10.1007/s10623-012-9691-5.
[14]	D. Silva and F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, IT-54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.
[15]	A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160. doi: 10.3934/amc.2013.7.147.
[16]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theory, IT-59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.
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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. Inf. Theory, IT-46 (2000), 1204-1216. doi: 10.1109/18.850663.
[2]	M. Braun, T. Etzion, P. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems,, preprint, ().
[3]	S. El-Zanati, H. Jordon, G. Seelinger, P. Sissokho and L. Spence, The maximum size of a partial $3$-spread in a finite vector space over $\mathbb F_2$, Des. Codes Crypt., 54 (2010), 101-107. doi: 10.1007/s10623-009-9311-1.
[4]	A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, Hungary, 2010, 1811-1814.
[5]	T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, IT-55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.
[6]	T. Etzion and A. Vardy, Error-correcting codes in projective geometry, IEEE Trans. Inf. Theory, IT-57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232.
[7]	T. Etzion and A. Vardy, On $q$-analogs of Steiner systems and covering designs, Adv. Math. Commun., 5 (2011), 161-176. doi: 10.3934/amc.2011.5.161.
[8]	E. M. Gabidulin, Theory of codes with maximal rank distance, Probl. Inf. Transm., 21 (1985), 1-12.
[9]	E. M. Gabidulin, N. I. Pilipchuk and M. Bossert, Decoding of random network codes, Probl. Inf. Trans. (Engl. Transl.), 46 (2010), 300-320. doi: 10.1134/S0032946010040034.
[10]	A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in Proc. 12th IMA Conf. Crypt. Coding, Cirencester, 2009, 1-21. doi: 10.1007/978-3-642-10868-6_1.
[11]	R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, IT-54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.
[12]	A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science (eds. J. Calmet, W. Geiselmann and J. Müller-Quade), Springer, Berlin, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4.
[13]	J. Rosenthal and A.-L. Trautmann, A complete characterization of irreducible cyclic orbit codes and their Plücker embedding, Des. Codes Crypt., 66 (2013), 275-289. doi: 10.1007/s10623-012-9691-5.
[14]	D. Silva and F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, IT-54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.
[15]	A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160. doi: 10.3934/amc.2013.7.147.
[16]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theory, IT-59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.
[17]	S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Crypt., 50 (2009), 163-172. doi: 10.1007/s10623-008-9221-7.

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