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

Abstract

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

“…Most of them provide constant dimension codes which contain a lifted MRD code as a subcode. Another type of constructions includes orbit or cyclic codes [5], [11], [18]. In [4], an upper bound on the cardinality of codes which contain a lifted MRD code was presented for some sets of parameters.…”

Message encoding and retrieval for spread and cyclic orbit codes

“…The usage of normalized representatives of points is a common concept there. Orbit codes [38] in G q (k, n) are defined to be orbits of a subgroup of the general linear group GL n := {A ∈ F n×n q | rank(A) = n} of order n over F q : Definition 14. Let U ∈ G q (k, n) and G be a subgroup of GL n .…”

Message encoding for spread and orbit codes

“…Orbit codes [23] in G q (k, n) are defined to be orbits of a subgroup of the general linear group GL n of order n over F q . They can be seen as the analogs of linear codes in classical block coding and their structure can be used for an easy computation of the minimum distance of a code and for decoding algorithms (e.g.…”

“…Hence, for our purposes, it remains to investigate when a primitive cyclic orbit spread code is a Desarguesian spread. This can be done by using the algorithm of [5], or by using the following results: Desarguesian spreads are always orbit codes [23], and two orbit codes are linearly isometric if and only if their generating groups are conjugates [14]. Furthermore, if one of two conjugate groups is cyclic, also the other one is cyclic and there exist two respective generator matrices of the two groups, that are similar.…”