Designs, Codes and Cryptography

, Volume 66, Issue 1–3, pp 275–289 | Cite as

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

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.

Keywords

Network coding Constant dimension codes Grassmannian Plücker embedding Projective space General linear group 

Mathematics Subject Classification

11T71 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of ZurichZurichSwitzerland

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