Abstract
Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Plücker embedding of these codes and show how the orbit structure is preserved in the embedding.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Ahlswede R., Cai N., Li S.-Y.R., Yeung R.W.: Network information flow. IEEE Trans. Inf. Theor. 46, 1204–1216 (2000)
Elsenhans A., Kohnert A., Wassermann A.: Construction of codes for network coding. In: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems—MTNS, pp. 1811–1814. Budapest (2010).
Etzion T., Silberstein N.: Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams. IEEE Trans. Inf. Theor. 55(7), 2909–2919 (2009)
Hirschfeld J.W.P.: Projective Geometries Over Finite Fields. Oxford Mathematical Monographs 2nd edn. The Clarendon Press Oxford University Press, New York (1998)
Hodge W.V.D., Pedoe D.: Methods of Algebraic Geometry, Vol. II. Cambridge University Press (1952).
Kohnert A., Kurz S.: Construction of large constant dimension codes with a prescribed minimum distance. In: Jacques C., Willi G., Müller-Quade Jörn (eds.), MMICS, vol. 5393 of Lecture Notes in Computer Science. Springer, pp. 31–42 (2008).
Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theor. 54(8), 3579–3591 (2008)
Lidl R., Niederreiter H.: Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge London (1986)
Manganiello F., Gorla E., Rosenthal J.: Spread codes and spread decoding in network coding. In: Proceedings of the 2008 IEEE International Symposium on Information Theory, pp. 851–855. Toronto, Canada (2008).
Manganiello F., Trautmann A.-L., Rosenthal J.: On conjugacy classes of subgroups of the general linear group and cyclic orbit codes. In: Proceedings of the 2011 IEEE International Symposium on Information Theory, pp. 1916–1920, 31, 5 Aug (2011).
Manganiello F., Trautmann A.-L.: Spread decoding in extension fields. arXiv:1108.5881v1 [cs.IT], (2011).
Silva D., Kschischang F.R., Kötter R.: A rank-metric approach to error control in random network coding. Proc. IEEE Int. Symp. Inf. Theor. 54(9), 3951–3967 (2008)
Trautmann A.-L., Manganiello F., Rosenthal J.: Orbit codes—a new concept in the area of network coding. In: Information Theory Workshop (ITW), IEEE, pp. 1–4, Dublin August (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
Rights and permissions
About this article
Cite this article
Rosenthal, J., Trautmann, AL. A complete characterization of irreducible cyclic orbit codes and their Plücker embedding. Des. Codes Cryptogr. 66, 275–289 (2013). https://doi.org/10.1007/s10623-012-9691-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9691-5
Keywords
- Network coding
- Constant dimension codes
- Grassmannian
- Plücker embedding
- Projective space
- General linear group