Cyclic Orbit Codes

In network coding a constant dimension code consists of a set of k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of F_q^n. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper we give a classification of cyclic orbit codes and propose a decoding procedure for a particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor

On conjugacy classes of subgroups of the general linear group and cyclic orbit codes

Orbit codes are a family of codes employable for communications on a random linear network coding channel. The paper focuses on the classification of these codes. We start by classifying the conjugacy classes of cyclic subgroups of the general linear group. As a result, we are able to focus the study of cyclic orbit codes to a restricted family of them.Comment: 5 pages; Submitted to IEEE International Symposium on Information Theory (ISIT) 201

Weight Distributions, Automorphisms, and Isometries of Cyclic Orbit Codes

Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup Fqn* on an Fq-subspace U of Fqn. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference space in the code. My dissertation investigates the structure of the weight distribution for cyclic orbit codes. We show that for full-length orbit codes with maximal possible distance the weight distribution depends only on q,n and the dimension of U. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the weight distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. We also briefly address the weight distribution of a union of full-length orbit codes with maximum distance. A related problem is to find the automorphism group of a cyclic orbit code, which plays a role in determining the isometry classes of the set of all cyclic orbit codes. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of Fqn. We then generalize to orbits under the normalizer of the Singer subgroup, although in this setup there is a remaining exceptional case. Finally, we can characterize linear isometries between such codes

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

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.Comment: submitted to Designs, Codes and Cryptograph

Pl\"ucker Embedding of Cyclic Orbit Codes

Cyclic orbit codes are a family of constant dimension codes used for random network coding. We investigate the Pl\"ucker embedding of these codes and show how to efficiently compute the Grassmann coordinates of the code words.Comment: to appear in Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems 2012, Melbourne, Australi

A Complete Characterization of Irreducible Cyclic Orbit Codes

We give a complete list of orbit codes that are generated by an irreducible cyclic group, i.e. an irreducible group having one generator. We derive some of the basic properties of these codes such as the cardinality and the minimum distance.Comment: in Proceedings of The Seventh International Workshop on Coding and Cryptography 2011 April 11-15 2011, Paris, Franc