American Institute of Mathematical Sciences

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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

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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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.  Google Scholar

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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.  Google Scholar

[2]

M. Braun, T. Etzion, P. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems,, preprint, ().   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. M. Gabidulin, Theory of codes with maximal rank distance, Probl. Inf. Transm., 21 (1985), 1-12.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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