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

Silberstein, Natalia; Trautmann, Anna-Lena

doi:10.1109/tit.2015.2435743

Cited by 78 publications

(69 citation statements)

References 24 publications

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“…We refer some known results about general lower bounds for A q (n, d, k) to [6,7,19,21]. Many CDC's from the multilevel construction based on echelon-Ferrers diagrams have been given.…”

Section: Previous Constructionsmentioning

confidence: 99%

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New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

Chen

Weng

et al. 2020

IEEE Trans. Inform. Theory

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A basic problem for the constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of a set of k-dimensional subspaces in F n q such that the subspace distance satisfies d(U, V ) = 2k − 2 dim(U ∩ V ) ≥ d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximal rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem of the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A q (n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

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“…We refer some known results about general lower bounds for A q (n, d, k) to [6,7,19,21]. Many CDC's from the multilevel construction based on echelon-Ferrers diagrams have been given.…”

Section: Previous Constructionsmentioning

confidence: 99%

“…Many CDC's from the multilevel construction based on echelon-Ferrers diagrams have been given. For example it was proved in [19] that when q 2 + q + 1 ≥ n − k 2 +k−6 2 and in some other cases (see [19])…”

Section: Previous Constructionsmentioning

confidence: 99%

New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

Chen

Weng

et al. 2020

IEEE Trans. Inform. Theory

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“…We do not intend to give any specific construction as this is a topic for another research work (see [12]). Note, that most constructions for λ = 1 can be generalized for larger λ if k ≤ n − k. As for λ = 1, also for larger λ it is not difficult to design constructions based on projective geometry [12], on Ferrers diagrams [10], [13], [35], and on rank-metric codes [36].…”

Section: Analysis Of the Related Codesmentioning

confidence: 99%

Grassmannian Codes with New Distance Measures for Network Coding

Etzion

Zhang

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this work we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts it is proved that codes in the Hamming scheme form a subfamily of the Grassmannian codes.

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“…Five constructions for FDRM codes are presented in this paper. All known constructions from [1,4,5,10,13,21,26] cannot produce optimal FDRM codes obtained from Examples 2.4, 2.5, 2.10, 2.11, 3.4 and 4.1. For future research, we suggest the following problems.…”

Section: Discussionmentioning

confidence: 96%

Several classes of optimal Ferrers diagram rank-metric codes

Liu

Chang

Feng

2019

Linear Algebra and its Applications

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Five constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways of representing elements of a finite field F q m (vector representation and matrix representation). The last one is based on Ferrers diagram Kronecker products. Each of these constructions produces optimal codes with different diagrams and parameters for which no optimal construction was known before.

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

Cited by 78 publications

References 24 publications

New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

Grassmannian Codes with New Distance Measures for Network Coding

Several classes of optimal Ferrers diagram rank-metric codes

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