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Efficient Decoding of Gabidulin Codes over Galois Rings [article]

Sven Puchinger and Julian Renner and Antonia Wachter-Zeh and Jens Zumbrägel
2021 arXiv   pre-print
This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity.  ...  This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.  ...  A NEW DECODER FOR GABIDULIN CODES OVER GALOIS RINGS In this section, we propose a new decoding algorithm for Gabidulin codes over rings with quadratic complexity in the code length.  ... 
arXiv:2102.02157v1 fatcat:ek7z746aojadndtjpkmtgqf5aa

Efficient Decoding of Gabidulin Codes over Galois Rings

Sven Puchinger, Julian Renner, Antonia Wachter-Zeh, Jens Zumbrage
2021 2021 IEEE International Symposium on Information Theory (ISIT)  
This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity.  ...  This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.  ...  A NEW DECODER FOR GABIDULIN CODES OVER GALOIS RINGS In this section, we propose a new decoding algorithm for Gabidulin codes over rings with quadratic complexity in the code length.  ... 
doi:10.1109/isit45174.2021.9517904 fatcat:elshj3pk45ftvmfpw5l3uy6niu

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.  ...  Since most applications of LRMR deal with matrices over infinite fields, we use the recently introduced generalization of Gabidulin codes to fields of characterstic zero.  ...  Gabidulin codes in characteristic zero are defined using θ-polynomials which are a generalization of linearized polynomials known from the definition of Gabidulin codes over finite fields.  ... 
arXiv:1604.04397v2 fatcat:pucnc77xdve57admuavbjb5pcq

A quadratic Welch-Berlekamp algorithm to decode generalized Gabidulin codes, and some variants

Gwezheneg Robert
2016 2016 IEEE International Symposium on Information Theory (ISIT)  
A quadratic Welch-Berlekamp algorithm to decode generalized Gabidulin codes, and some variants.  ...  DECODING BY RECONSTRUCTION The reconstruction problem and its adaptation to the decoding of Gabidulin codes is described in [6] . The method is also valid for Gabidulin codes over infinite fields..  ...  A divisionfree variant enables the use of this algorithm for codes over integer rings [11] .  ... 
doi:10.1109/isit.2016.7541761 dblp:conf/isit/Robert16 fatcat:67u5ruhtarflnaem2sz6ohu7ty

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.  ...  Gabidulin Codes Gabidulin codes were originally defined by [3] , [2] , [4] over finite fields.  ... 
arXiv:1601.05205v2 fatcat:alqqhr7mp5a23a4x6zvcp3zi4q

On the Rank Decoding Problem Over Finite Principal Ideal Rings [article]

Hervé Tale Kalachi, Hermann Tchatchiem Kamche
2022 arXiv   pre-print
But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings  ...  We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields.  ...  This work was followed by [26] where the authors provide an iterative decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity in the code length.  ... 
arXiv:2106.11569v3 fatcat:n6hoaffotrfgbid4lsrmdlz6pm

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 derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm.  ...  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 derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm.  ...  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

Generalized Gabidulin codes over fields of any characteristic [article]

Daniel Augot, Pierre Loidreau, Gwezheneg Robert
2017 arXiv   pre-print
Under these conditions, we generalize Gabidulin codes and provide a decoding algorithm which decode both errors and erasures. Then, we focus on codes over integer rings and how to decode them.  ...  For this, we study the reduction of generalized Gabidulin codes over number ideals codes modulo a prime ideal, and show they are classical Gabidulin codes.  ...  Gabidulin code over a finite field.  ... 
arXiv:1703.09125v1 fatcat:tshniykevzae7go24gj62ggvqy

Rank-Metric Codes and Their Applications [article]

Hannes Bartz, Lukas Holzbaur, Hedongliang Liu, Sven Puchinger, Julian Renner, Antonia Wachter-Zeh
2022 arXiv   pre-print
In code-based cryptography, the hardness of the corresponding generic decoding problem can lead to systems with reduced public-key size.  ...  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.  ...  Acknowledgements The work of L. Holzbaur and A. Wachter  ... 
arXiv:2203.12384v1 fatcat:brysqya7ynem5i25adxfwf5vl4

Page 3654 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews  
We extend these results to linear codes over the Galois ring Z4 and apply the theory to Z4-codes of length 24.  ...  A decoding procedure for our codes is also considered.” 99e:94042 94B05 Gabidulin, Ernst M. [Gabidulin, E. M.]  ... 

Low-Rank Parity-Check Codes over Galois Rings [article]

Julian Renner, Alessandro Neri, Sven Puchinger
2020 Designs, Codes and Cryptography   accepted
Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide class of finite commutative rings.  ...  We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best  ...  Acknowledgment The work of J. Renner was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 801434). A.  ... 
doi:10.1007/s10623-020-00825-9 pmid:33603280 pmcid:PMC7870781 arXiv:2006.10588v2 fatcat:bafdjqtbyvc2xgff5czoyuxe74

Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes [article]

Daniel Augot, Alain Couvreur, Julien Lavauzelle, Alessandro Neri
2020 arXiv   pre-print
This paper extends the study of rank-metric codes in extension fields L equipped with an arbitrary Galois group G = Gal(L/K).  ...  We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes.  ...  Acknowledgements The authors would like to thank the organizers of Dagstuhl seminar no. 18511 "Algebraic Coding Theory for Networks, Storage, and Security" where was initiated this research project.  ... 
arXiv:2006.14489v1 fatcat:y3m5ati3dvhmdbdbydtbcpaqdm

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric [article]

Hannes Bartz and Thomas Jerkovits and Sven Puchinger and Johan Rosenkilde
2021 arXiv   pre-print
We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized  ...  skew polynomial rings.  ...  Hence, any efficient decoder for skew Reed-Solomon codes in the skew metric is also an efficient decoder for linearized Reed-Solomon codes in the sum-rank metric.  ... 
arXiv:2005.09916v2 fatcat:sqtkw73frraqhfiol7evekjcwy

Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings and its Application in Coding Theory [article]

Hannes Bartz and Thomas Jerkovits and Johan Rosenkilde
2022 arXiv   pre-print
We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed-Solomon codes  ...  This matches the previous best speeds for these tasks, which were obtained by top-down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial  ...  Interpolation-Based Decoding of Interleaved Gabidulin Codes Interleaved Gabidulin codes are rank-metric codes that are obtained by the Cartesian product of ordinary Gabidulin codes [19] and which allow  ... 
arXiv:2207.01319v1 fatcat:66mqx2nd5jgdbkuonm5u4kzgs4
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