Skip to main content

A complete characterization of irreducible cyclic orbit codes and their Plücker embedding

Designs, Codes and Cryptography volume 66, pages 275–289 (2013)Cite this article

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, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Ahlswede R., Cai N., Li S.-Y.R., Yeung R.W.: Network information flow. IEEE Trans. Inf. Theor. 46, 1204–1216 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  2. 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).

  3. 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)

    MathSciNet  Article  Google Scholar 

  4. Hirschfeld J.W.P.: Projective Geometries Over Finite Fields. Oxford Mathematical Monographs 2nd edn. The Clarendon Press Oxford University Press, New York (1998)

    Google Scholar 

  5. Hodge W.V.D., Pedoe D.: Methods of Algebraic Geometry, Vol. II. Cambridge University Press (1952).

  6. 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).

  7. Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theor. 54(8), 3579–3591 (2008)

    Article  Google Scholar 

  8. Lidl R., Niederreiter H.: Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge London (1986)

    MATH  Google Scholar 

  9. 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).

  10. 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).

  11. Manganiello F., Trautmann A.-L.: Spread decoding in extension fields. arXiv:1108.5881v1 [cs.IT], (2011).

  12. 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)

    Article  Google Scholar 

  13. 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).

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Zurich, Zurich, Switzerland

    Joachim Rosenthal & Anna-Lena Trautmann

Authors
  1. Joachim Rosenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna-Lena Trautmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna-Lena Trautmann.

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

Reprints 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

Download citation

  • 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

Mathematics Subject Classification

  • 11T71