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Rank metric and Gabidulin codes in characteristic zero [article]

Gwezheneg Robert, Pierre Loidreau, Daniel Augot
2013 arXiv   pre-print
We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group.  ...  We also provide various definitions for the rank-metric.  ...  Lu and Kumar [7] used an original approach by transforming optimal codes in rank metric over finite fields, such as Gabidulin codes, into optimal codes for space-time coding over different types of constellations  ... 
arXiv:1305.4047v1 fatcat:by3ecimsf5cfdbquvsfljkz4gq

Rank metric and Gabidulin codes in characteristic zero

Daniel Augot, Pierre Loidreau, Gwezheneg Robert
2013 2013 IEEE International Symposium on Information Theory  
We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group.  ...  We also provide various definitions for the rank-metric.  ...  Lu and Kumar [7] used an original approach by transforming optimal codes in rank metric over finite fields, such as Gabidulin codes, into optimal codes for space-time coding over different types of constellations  ... 
doi:10.1109/isit.2013.6620278 dblp:conf/isit/AugotLR13 fatcat:z6jey3me2ngkxg42nalru7wwlu

An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero [article]

Sven Müelich and Sven Puchinger and David Mödinger and Martin Bossert
2016 arXiv   pre-print
Gabidulin codes, originally defined over finite fields, are an important class of rank metric codes with various applications.  ...  We propose a new application of Gabidulin codes over infinite fields: low-rank matrix recovery.  ...  Index Terms-Gabidulin Codes, Characteristic Zero, Rank Metric, Decoding, Matrix Recovery I. MOTIVATION Finding a matrix of minimal rank is a problem which occurs in different scenarios.  ... 
arXiv:1601.05205v2 fatcat:alqqhr7mp5a23a4x6zvcp3zi4q

Low-Rank Matrix Recovery using Gabidulin Codes in Characteristic Zero [article]

Sven Müelich, Sven Puchinger, Martin Bossert
2016 arXiv   pre-print
We show that LRMR can be reduced to decoding of Gabidulin codes and discuss which field extensions can be used in the code construction.  ...  We present a new approach on low-rank matrix recovery (LRMR) based on Gabidulin Codes.  ...  Gabidulin Codes in Characteristic Zero Gabidulin codes over finite fields were introduced in [7, 8, 9] , a comprehensive overview is given in [10] .  ... 
arXiv:1604.04397v2 fatcat:pucnc77xdve57admuavbjb5pcq

Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero [article]

Hikmet Yildiz, Netanel Raviv, Babak Hassibi
2020 arXiv   pre-print
In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero.  ...  Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic.  ...  PROBLEM SETUP In this section we will first provide a brief background on cyclic Galois extensions. Then, we will define rank metric codes and Gabidulin codes.  ... 
arXiv:2004.14430v1 fatcat:u7c5sy5vdjf2zdoonodtbuvcda

Constructions of optimal rank-metric codes from automorphisms of rational function fields [article]

Rakhi Pratihar, Tovohery Hajatiana Randrianarisoa
2021 arXiv   pre-print
We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes.  ...  We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.  ...  More recently, Augot et al. have extended the notion of rank metric to arbitrary field extensions, possibly of characteristic zero in [4] .  ... 
arXiv:1907.05508v4 fatcat:3c5nhg7mwvhmrdcpxov5rhbruy

Constructions of optimal rank-metric codes from automorphisms of rational function fields

Rakhi Pratihar, Tovohery Hajatiana Randrianarisoa
2022 Advances in Mathematics of Communications  
<p style='text-indent:20px;'>We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes.  ...  We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.</p>  ...  No. 2061641156) and by Grant 280731 from the Research Council of Norway.  ... 
doi:10.3934/amc.2022034 fatcat:p475vliwpjda3jk74x4t2kbwwi

Rank-metric LCD codes [article]

Xiusheng Liu, Hualu Liu
2017 arXiv   pre-print
In this paper, we investigate the rank-metric codes which are proposed by Delsarte and Gabidulin to be complementary dual codes.  ...  We point out the relationship between Delsarte complementary dual codes and Gabidulin complementary dual codes.  ...  The necessary background materials of LCD and Delsarte rank-metric codes are given in Section 2. In Section 3, we investigate the relationship between Delsarte LCD codes and Gabidulin LCD codes.  ... 
arXiv:1708.00997v1 fatcat:uwie5pbvbzf25gzj7j2kifi22m

On the list decodability of Rank Metric codes [article]

Rocco Trombetti, Ferdinando Zullo
2020 arXiv   pre-print
In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and  ...  Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove that any rank metric code of F_q^m^n of order q^kn with n dividing m,  ...  In fact, this will yield to a generalization of Theorem 16 and Corollary 17 to any rank-metric code of Gabidulin index two.  ... 
arXiv:1907.01289v2 fatcat:oab2kb3tnratffryrkwzzobzki

Automorphism groups of Gabidulin-like codes [article]

Dirk Liebhold, Gabriele Nebe
2016 arXiv   pre-print
This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.  ...  . , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank distance code in the space of f times m matrices over k of dimension fd  ...  In the special case of classical Gabidulin codes of full length m = ℓ these groups have already been determined in [10] and [14] . 2 Rank metric codes. Let k be any field, ℓ, m ∈ N.  ... 
arXiv:1603.09565v1 fatcat:uf3xib5u6rh4pjhb4inwmcqhne

On deep-holes of Gabidulin codes [article]

Weijun Fang, Li-Ping Wang, Daqing Wan
2018 arXiv   pre-print
In this paper, we determine the covering radius and a class of deep holes for Gabidulin codes with both rank metric and Hamming metric.  ...  Moreover, we give a necessary and sufficient condition for deciding whether a word is not a deep hole for Gabidulin codes, by which we study the error distance of a special class of words to certain Gabidulin  ...  Conclusions In this paper, we study deep holes of Gabidulin codes in both Hamming metric and rank metric.  ... 
arXiv:1711.11202v3 fatcat:rdqq6ifuu5e4tgyw4dhnif7jty

Self-duality of generalized twisted Gabidulin codes

Kamil Otal, Ferruh Özbudak, Wolfgang Willems
2018 Advances in Mathematics of Communications  
Self-duality of Gabidulin codes was investigated in [10] and the authors provided an if and only if condition for a Gabidulin code to be equivalent to a self-dual maximum rank distance (MRD) code.  ...  We observe that the condition presented in [10] still holds for generalized Gabidulin codes (an intermediate family between Gabidulin codes and generalized twisted Gabidulin codes).  ...  The third author would like to thank METU for its warm-hearted hospitality and generous financial support during his stay at the University in January 2016.  ... 
doi:10.3934/amc.2018042 fatcat:qubwpwhs6rb5rhplavnjxl3xxe

On the list decodability of rank-metric codes containing Gabidulin codes [article]

Paolo Santonastaso, Ferdinando Zullo
2021 arXiv   pre-print
We detect several families of rank-metric codes containing a generalized Gabidulin code as subcode which cannot be efficiently list decoded for any radius large enough and families of rank-metric codes  ...  In this paper, we provide bounds on the list size of rank-metric codes containing generalized Gabidulin codes in order to determine whether or not a polynomial-time list decoding algorithm exists.  ...  Acknowledgements This research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA -INdAM).  ... 
arXiv:2103.07547v1 fatcat:qc62d3rcivhrvirmhvgwu3m4wq

Rank-Metric Codes for Priority Encoding Transmission with Network Coding

Danilo Silva, Frank R. Kschischang
2007 2007 10th Canadian Workshop on Information Theory (CWIT)  
In this paper, a PET system based on rank-metric codes is proposed. This system can provide strict combinatorial guarantees of recovery.  ...  So far, the only solution proposed relies on the rateless feature of network coding and is therefore not suitable for delay-constrained applications.  ...  Gabidulin codes have maximal distance in the rank metric and are the analogous of Reed-Solomon codes, which have maximal distance in the Hamming metric.  ... 
doi:10.1109/cwit.2007.375706 fatcat:4qugqsspxvb7ndyivksqnuk7ki

Fast Encoding and Decoding of Gabidulin Codes [article]

Danilo Silva, Frank R. Kschischang
2009 arXiv   pre-print
Gabidulin codes are the rank-metric analogs of Reed-Solomon codes and have a major role in practical error control for network coding.  ...  In addition, a new decoding algorithm is proposed based on a transform-domain approach. Together, these represent the fastest known algorithms for encoding and decoding Gabidulin codes.  ...  INTRODUCTION Gabidulin codes [1] are optimal codes for the rank metric that are closely related to Reed-Solomon codes.  ... 
arXiv:0901.2483v2 fatcat:2h5azdvx3re3tazrzwpknpsd5y
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