### Message encoding for spread and orbit codes

- In Proceedings of the 2014 IEEE International Symposium on Information Theory (ISIT
, 2014

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### On the Decoder Error Probability of Rank Metric Codes and Constant-Dimension Codes

, 2008

"... Rank metric codes can either be used as such for error correction in data storage equipments, or be lifted into constant-dimension codes (CDCs) and thus be used for error correction in random network coding. This paper investigates the decoder error probability (DEP) of rank metric codes and CDCs. W ..."

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Rank metric codes can either be used as such for error correction in data storage equipments, or be lifted into constant-dimension codes (CDCs) and thus be used for error correction in random network coding. This paper investigates the decoder error probability (DEP) of rank metric codes and CDCs. We first study the DEP of rank metric codes using a bounded rank distance decoder. We derive asymptotically tight upper bounds on the DEP of rank metric codes used over an equal row or an equal column space channel, and we determine the exact DEP of maximum rank distance codes used over an equal row space channel. We then investigate the DEP of CDCs using a bounded subspace distance or a bounded modified subspace distance decoder over a symmetric operator channel. We first determine some fundamental properties of the subspace and the modified subspace metrics. Using these properties, we derive asymptotically tight upper bounds on the DEP of any CDC obtained by lifting a rank metric code using either a bounded subspace distance or a bounded modified subspace distance decoder.

### Enumerative Encoding in the Grassmannian Space

, 903

"... Abstract — Codes in the Grassmannian space have found recently application in network coding. Representation of k-dimensional subspaces of F n q has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different repr ..."

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Abstract — Codes in the Grassmannian space have found recently application in network coding. Representation of k-dimensional subspaces of F n q has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different representations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on Ferrers diagrams representation of subspaces; and another is based on identifying vector and reduced row echelon form representation of subspaces. A third method which combine the previous two is more efficient than the other two enumerative encodings. I.

### Gray Codes and Enumerative Coding for Vector Spaces

"... Abstract—Gray codes for vector spaces are considered in two graphs: the Grassmann graph, and the projective-space graph, both of which have recently found applications in network coding. For the Grassmann graph, constructions of cyclic optimal codes are given for all parameters. As for the projectiv ..."

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Abstract—Gray codes for vector spaces are considered in two graphs: the Grassmann graph, and the projective-space graph, both of which have recently found applications in network coding. For the Grassmann graph, constructions of cyclic optimal codes are given for all parameters. As for the projective-space graph, two constructions for specific parameters are provided, as well some nonexistence results. Furthermore, encoding and decoding algorithms are given for the Grassmannian Gray code, which induce an enumerative-coding scheme. The computational complexity of the algorithms is at least as low as known schemes, and for certain parameter ranges, the new scheme outperforms previously known ones. Index Terms—Enumerative coding, Grassmannian, Gray codes, projective-space graph.

### Message Encoding and Retrieval for Spread and Cyclic Orbit Codes

"... Spread codes and cyclic orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability, transmission rate and decoding methods, but the question of how to encode and retrieve messages has not been investigated. In this ..."

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Spread codes and cyclic orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability, transmission rate and decoding methods, but the question of how to encode and retrieve messages has not been investigated. In this work we show how a message set of consecutive integers can be encoded and retrieved for these two code families. Index Terms Message encoding, network coding, constant dimension codes, subspace codes, Grassmannian, enumerative coding, orbit

### Optimal Ferrers Diagram Rank-Metric Codes

, 2014

"... Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing o ..."

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Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters.