A Complete Characterization of Irreducible Cyclic Orbit Codes

Constant dimension codes are subsets of the finite Grassmann variety. The study of constant dimension codes with good distances have been central in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are characterized that the elements of the code can be viewed as the orbit under a group action. The paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. an irreducible group having one generator. 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.