A Complete Characterization of Irreducible Cyclic Orbit Codes and their Pl\"ucker Embedding

TL;DR

This paper fully characterizes irreducible cyclic orbit codes within constant dimension codes, exploring their properties and how their structure is preserved under Pl"ucker embedding, advancing understanding in network coding theory.

Contribution

It provides a complete characterization of irreducible cyclic orbit codes and analyzes their properties and embeddings, which was previously not fully understood.

Findings

Cardinality and minimum distance derived from vector space and extension field isomorphism

Orbit structure preserved under Pl"ucker embedding

Complete classification of irreducible cyclic orbit codes

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 Pl\"ucker embedding of these codes and show how the orbit structure is preserved in the embedding.