Abstract
Given a finite field \(\mathbb {F}_{q}\), a constant dimension code is a set of k-dimensional subspaces of \(\mathbb {F}_{q}^{n}\). Orbit codes are constant dimension codes which are defined as orbits when the action of a subgroup of the general linear group on the set of all subspaces of \(\mathbb {F}_{q}^{n}\) is considered. In this paper we present a construction of an Abelian non-cyclic orbit code whose minimum subspace distance is maximal.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bardestani, F., Iranmanesh, A.: Cyclic orbit codes with the normalizer of a Singer subgroup. J. Sci. Islamic Republic of Iran 26(1), 49–55 (2015)
Bartoli, D., Pavese, F.: A note on equidistant subspace codes. Discrete Appl. Math. 198, 291–296 (2016)
Ben-Sasson, E., Etzion, T., Gabizon, A., Raviv, N.: Subspace polynomials and cyclic subspace codes. IEEE Trans. Inf. Theory 62(3), 1157–1165 (2016)
Cossidente, A., Pavese, F.: On subspace codes. Des. Codes Crypt. 78(2), 527–531 (2016)
Etzion, T., Vardy, A.: Error-correcting codes in projective space. IEEE Trans. Inf. Theory 57(2), 1165–1173 (2011)
Ghatak, A.: Construction of Singer subgroup orbit codes based on cyclic difference sets. In: Proceedings of the Twentieth National Conference on Communications (NCC 2014), pp. 1–4. IEEE, Kanpur, India, February 2014
Gluesing-Luerssen, H., Morrison, K., Troha, C.: Cyclic orbit codes and stabilizer subfields. Adv. Math. Commun. 9(2), 177–197 (2015)
Gluesing-Luerssen, H., Troha, C.: Construction of subspace codes through linkage. Adv. Math. Commun. 10(3), 525–540 (2016)
Gorla, E.G., Ravagnani, A.: Equidistant subspace codes. Linear Algebra Appl. 490, 48–65 (2016)
Ho, T., Koetter, R., Médard, M., Karger, D.R., Effros, M.: The benefits of coding over routing in a randomized setting. In: Proceedings of the 2003 IEEE International Symposium on Information Theory (ISIT 2003), p. 442. IEEE, Yokohama, Japan, June/July 2003
Koetter, R., Kschischang, F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008)
Rosenthal, J., Trautmann, A.L.: A complete characterization of irreducible cyclic orbit codes and their Plücker embedding. Des. Codes Crypt. 66, 275–289 (2013)
Silberstein, N., Trautmann, A.L.: New lower bounds for constant dimension codes. In: Proceedings of the 2013 IEEE International Symposium on Information Theory (ISIT 2013), pp. 514–518. IEEE, Istanbul, July 2013
Slepian, D.: Group codes for the Gaussian channel. Bell Syst. Tech. J. 47(4), 575–602 (1968)
Trautmann, A.L.: Isometry and automorphisms of constant dimension codes. Adv. Math. Commun. 7(2), 147–160 (2013)
Trautmann, A.L., Manganiello, F., Braun, M., Rosenthal, J.: Cyclic orbit codes. IEEE Trans. Inf. Theory 59(11), 7386–7404 (2013)
Trautmann, A.L., Manganiello, F., Rosenthal, J.: Orbit codes - a new concept in the area of network coding. In: Proceedings of the 2010 IEEE Information Theory Workshop (ITW 2010). IEEE, Dublin, Ireland, August 2010
Acknowledgements
The first author was supported by grants MIMECO MTM2015-68805-REDT and MTM2015-69138-REDT.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Climent, JJ., Requena, V., Soler-Escrivà, X. (2017). A Construction of Orbit Codes. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-66278-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66277-0
Online ISBN: 978-3-319-66278-7
eBook Packages: Computer ScienceComputer Science (R0)