Plücker Embedding of Cyclic Orbit Codes

Cyclic orbit codes are a family of constant dimension codes used for random network coding. We investigate the Plücker embedding of these codes and show how to efficiently compute the Grassmann coordinates of the code words. 1. Introduction. In network coding one is looking at the transmission of information through a directed graph with possibly several senders and several receivers [1]. One can increase the throughput by linearly combining the information vectors at intermediate nodes of the network. If the underlying topology of the network is unknown we speak about random linear network coding. Since linear spaces are invariant under linear combinations, they are what is needed as codewords [5]. It is helpful (e.g. for decoding) to constrain oneself to subspaces of a fixed dimension, in which case we talk about constant dimension codes. The general linear group, consisting of all invertible transformations acts naturally on the set of all k-dimensional vector spaces, called the Grassmann variety. Orbits under this action are called orbit codes [10]. Orbit codes have useful algebraic structure and can be seen as subspace analogues of linear block codes in some sense [9]. One can describe the balls of subspace radius 2t in the Grassmann variety in its Plücker embedding. Such an algebraic description of the balls of radius 2t is potentially important if one is interested in an algebraic decoding algorithm for constant dimension codes. For instance, a list decoding algorithm requires the computation of all code words which lie in some ball around a received message word. In this work we characterize the Plücker embedding of orbit codes generated by cyclic subgroups of the general linear group. The case of irreducible cyclic subgroups has already been studied in [7]. This work generalizes and completes those results for general cyclic orbit codes. The paper is structured as follows: In Section 2 we give some preliminaries on random network coding and orbit codes in particular. Moreover, we define irreducible and completely reducible matrices and groups. Section 3 contains the main results of this work. We first investigate the balls around subspaces of radius 2t with respect to the Grassmann coordinates. Then we recall how to efficiently compute these coordinates of irreducible cyclic orbit codes and show how the same can be done for reducible cyclic orbit codes. For this we distinguish between completely reducible and non-completely reducible generating groups. In Section 4 we conclude this work.