Anna-Lena Trautmann

Cyclic Orbit Codes

A constant dimension code consists of a set of k-dimensional subspaces of \BBF qn. 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 \BBF qn. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper, we show how orbit codes can be seen as an analog of linear codes in the block coding case. We investigate how the structure of cyclic orbit codes can be utilized to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass of cyclic orbit codes.

Orbit codes — A new concept in the area of network coding

We introduce a new class of constant dimension codes called orbit codes. The basic properties of these codes are derived. It will be shown that many of the known families of constant dimension codes in the literature are actually orbit codes.

Subspace Codes Based on Graph Matchings, Ferrers Diagrams, and Pending Blocks

This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rank-metric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with minimum injection distance 2 or k - 1, where k is the constant dimension. Furthermore, we present a construction of new codes from old codes for any minimum distance. Then, we construct nonconstant dimension codes from these codes. Some examples of codes obtained by these constructions are the largest known codes for the given parameters.

A complete characterization of irreducible cyclic orbit codes and their Plücker 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 Plücker embedding of these codes and show how the orbit structure is preserved in the embedding.

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

Orbit codes are a family of codes applicable 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.

New lower bounds for constant dimension codes

This paper provides new constructive lower bounds for constant dimension codes, using Ferrers diagram rank metric codes and pending blocks. Constructions for two families of parameters of constant dimension codes are presented. The examples of codes obtained by these constructions are the largest known constant dimension codes for the given parameters.

Cross-packing lattices for the Rician fading channel

We introduce cross-packing lattices for Rician fading channels, motivated by a geometric interpretation stemming from the pairwise error probability analysis. We approximate the star bodies arising from the pairwise error probability analysis with n-dimensional crosses of radius t, consisting of 2nt + 1 unit cubes, for some positive integer t. We give a construction for a family of cross-packing lattices for all dimensions and any minimum cross distance 2t + 1. We show by simulations how our new cross-packing lattices perform compared to other known lattices over the Rician fading channel, for different values of the Rician K-factor.

Message encoding for spread and orbit codes

Spread codes and orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability and transmission rate, but the question of how to encode and retrieve messages has not been investigated. In this work we show how the message space can be chosen for a given code and how message encoding and retrieval can be done.Spread codes and orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability and transmission rate, but the question of how to encode messages has not been investigated. In this work we show how the message space can be chosen for a given code and how message en- and decoding can be done.

Iterative list-decoding of Gabidulin codes via Gröbner based interpolation

We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.