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Properties of codes in rank metric [article]

P. Loidreau
2006 arXiv   pre-print
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist.  ...  We show that there are quasi-perfect codes in rank metric over fields of characteristic 2.  ... 
arXiv:cs/0610057v1 fatcat:gita7e2qrna7vijtk7enzzgj6u

Properties of subspace subcodes of optimum codes in rank metric [article]

E. M. Gabidulin, P. Loidreau
2006 arXiv   pre-print
Maximum rank distance codes denoted MRD-codes are the equivalent in rank metric of MDS-codes.  ...  These codes can be seen as the analogs of Reed-Solomon codes (hereafter denoted RS-codes) for rank metric. In this paper their subspace subcodes are characterized.  ... 
arXiv:cs/0607108v1 fatcat:bkgw74qpr5have7ee2b2iun4qy

Rank-Metric Codes and Their Applications

Hannes Bartz, Lukas Holzbaur, Hedongliang Liu, Sven Puchinger, Julian Renner, Antonia Wachter-Zeh
2022 Foundations and Trends in Communications and Information Theory  
In distributed data storage, codes in the rank metric have been used repeatedly to construct codes with locality, and in coded caching, they have been employed for the placement of coded symbols.  ...  Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications.  ...  Several code constructions and basic properties of the rank metric show strong similarities to codes in the Hamming metric.  ... 
doi:10.1561/0100000119 fatcat:4wh62agk5rczpjlfr5bp3km67u

Properties of Codes with the Rank Metric [article]

Maximilien Gadouleau, Zhiyuan Yan
2006 arXiv   pre-print
In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular.  ...  For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic forms of these two bounds and the Singleton bound.  ...  In this paper, we study properties of rank metric codes in general and MRD codes in particular.  ... 
arXiv:cs/0610099v2 fatcat:eidkuwoh3beelecaers2qgoita

Bounds on Covering Codes with the Rank Metric [article]

Maximilien Gadouleau, Zhiyuan Yan
2009 arXiv   pre-print
In this paper, we investigate geometrical properties of the rank metric space and covering properties of rank metric codes.  ...  The geometrical properties and bounds proposed in this paper are significant to the design, decoding, and performance analysis of rank metric codes.  ...  In this paper, we focus on the geometrical properties of the rank metric space and covering properties of rank metric codes.  ... 
arXiv:0809.2968v2 fatcat:paoztsmg7naedgdopzbc3i6ro4

Application of rank metric codes in digital image watermarking

Pascal Lefèvre, Philippe Carré, Philippe Gaborit
2019 Signal processing. Image communication  
Using the well known Lattice QIM in the spatial domain, we propose to use a different kind of error correcting codes called rank metric codes.  ...  One can decode the original message if the matrix rank of the detected codeword is small enough. We propose a study to validate the concept of rank metric in watermarking applications.  ...  Rank metric codes In this section, we introduce rank metric codes definitions, properties and their applications in practice.  ... 
doi:10.1016/j.image.2018.12.015 fatcat:o47ai36h75dd7jjuawuuhio55u

Some Matroids Related to Sum-Rank Metric Codes [article]

Avijit Panja, Rakhi Pratihar, Tovohery Hajatiana Randrianarisoa
2022 arXiv   pre-print
We introduce the notion of sum-matroids and show its association with sum-rank metric codes.  ...  As a consequence, some results for sum-rank metric codes by Martínez-Peñas are generalized for sum-matroids. The sum-matroids generalize the notions of matroids and q-matroids.  ...  Sudhir Ghorpade for reading our work and giving many useful comments that helped us to make approapriate revisions in this version.  ... 
arXiv:1912.09984v4 fatcat:ftshc6gfqnh2potftalaoz52oq

Rank-Metric Codes and q-Polymatroids [article]

Elisa Gorla, Relinde Jurrius, Hiram H. López, Alberto Ravagnani
2019 arXiv   pre-print
We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal  ...  This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view.  ...  There is a natural way to associate a rank-metric code C to a vector rank-metric code C, in such a way that the metric properties are preserved.  ... 
arXiv:1803.10844v3 fatcat:qo33av7dgfaolf6civd7niscqm

A Geometric Approach to Rank Metric Codes and a Classification of Constant Weight Codes [article]

Tovohery Hajatiana Randrianarisoa
2020 arXiv   pre-print
In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes.  ...  We give a complete classification of constant rank weight code and we give their generalized rank weights.  ...  I also thank the anonymous reviewer who introduced me to the notion of linear sets.  ... 
arXiv:1907.04372v3 fatcat:p4ilmaxnvfbahgxg7mwg6peyca

Watermarking and Rank Metric Codes

Pascal Lefevre, Philippe Carre, Philippe Gaborit
2018 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)  
Using the well known Lattice QIM in the spatial domain, we analyze the interest of using a different kind of error correcting codes: rank metric codes.  ...  These codes are already used in communications for network coding but not used in the context of watermarking.  ...  Blue curve : Image error rate of LQIM method combined with a rank metric code in function of β.  ... 
doi:10.1109/icassp.2018.8462647 dblp:conf/icassp/LefevreCG18 fatcat:cjk547lyxvgepk2z2l2nps3bea

Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties [article]

Bryan Hernandez, Virgilio Sison
2015 arXiv   pre-print
The matrix codes are GF(p)-subspaces of the ring M_2(GF(p)) of 2 × 2 matrices over GF(p) on which the rank metric is applied, and are generated as one-sided proper principal ideals by idempotent elements  ...  Finally a weight function on subspace codes is analogously defined and its egalitarian property is examined.  ...  Section IV discusses the weight properties of rank metric codes, while Section V studies the weight properties of the associated subspace codes. II.  ... 
arXiv:1502.05808v1 fatcat:6ci2dbbfvbfm3kp4b4ypq5uuwe

Antipodal two-weight rank metric codes [article]

Rakhi Pratihar, Tovohery Hajatiana Randrianarisoa
2022 arXiv   pre-print
We consider the class of linear antipodal two-weight rank metric codes and discuss their properties and characterization in terms of t-spreads.  ...  As a consequence of our construction of two-weight rank metric codes, we get some explicit two-weight Hamming metric codes.  ...  In Section 3, we derive some properties of the ATW rank metric codes and t-spreads and establish a relation between these two objects.  ... 
arXiv:2208.07295v2 fatcat:lq6nxuoxkva3bc2qz275t7vvpa

Enhanced Algebraic Error Control for Random Linear Network Coding [article]

Zhiyuan Yan, Hongmei Xie
2012 arXiv   pre-print
subspace and rank metric codes.  ...  To support the first-tier decoding, we also investigate Hamming distance properties of three important families of subspace and rank metric codes, Gabidulin codes, Kotter--Kschischang codes, and Mahdavifar  ...  of subspace and rank metric codes.  ... 
arXiv:1205.0586v1 fatcat:ab57n5uo5jhuzhabzc266cvpqu

Decoder Error Probability of MRD Codes

Maximilien Gadouleau, Zhiyuan Yan
2006 2006 IEEE Information Theory Workshop - ITW '06 Chengdu  
Using this new concept, we derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable (MDS) codes.  ...  In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates.  ...  In [4] , space-time block codes with good rank properties have been proposed.  ... 
doi:10.1109/itw2.2006.323800 fatcat:4r7xnmiexzf6pfjkmrvzx2u3ny

On the Decoder Error Probability of Bounded Rank-Distance Decoders for Maximum RankDistance Codes

Maximilien Gadouleau, Zhiyuan Yan
2008 IEEE Transactions on Information Theory  
Index Terms-Bounded distance decoder, decoder error probability, rank metric codes.  ...  We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable codes.  ...  Due to another reviewer's comments, the precise expression in Lemma 8 replaced an upper bound in our original submission.  ... 
doi:10.1109/tit.2008.924697 fatcat:6gvoe2zrtnboplehdaehzrxwoq
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