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Evading Subspaces Over Large Fields and Explicit List-decodable Rank-metric Codes

Venkatesan Guruswami, Carol Wang, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We construct an explicit family of linear rank-metric codes over any field F h that enables efficient list decoding up to a fraction ρ of errors in the rank metric with a rate of 1 − ρ − ε, for any desired  ...  Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius  ...  We thank Antonia Wachter-Zeh for bringing to our attention the lack of a Johnson-type bound for list decoding rank-metric codes [24] .  ... 
doi:10.4230/lipics.approx-random.2014.748 dblp:conf/approx/GuruswamiW14 fatcat:3yjuvzbdj5boxfb4pgqkgcokma

Explicit rank-metric codes list-decodable with optimal redundancy [article]

Venkatesan Guruswami, Carol Wang
2013 arXiv   pre-print
We construct an explicit family of linear rank-metric codes over any field F_h that enables efficient list decoding up to a fraction ρ of errors in the rank metric with a rate of 1-ρ-ϵ, for any desired  ...  Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius  ...  Acknowledgment We thank Antonia Wachter-Zeh for bringing to our attention the lack of a Johnson-type bound for list decoding rank-metric codes [24] .  ... 
arXiv:1311.7084v2 fatcat:ycaqwsudzvdxxgsh52wvrts4fi

On defining generalized rank weights

Ruud Pellikaan, Relinde Jurrius
2017 Advances in Mathematics of Communications  
As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.  ...  We study rank metric codes over L, where L is a finite Galois extension of a field K. This is a generalization of the case where K = F_q and L = F_q^m of Gabidulin codes to arbitrary characteristic.  ...  Many notions in the theory for codes with the Hamming metric have an equivalent notion for codes with the rank metric.  ... 
doi:10.3934/amc.2017014 fatcat:3yocr7n45ra7fozeuetzaolzle

The interplay of different metrics for the construction of constant dimension codes [article]

Sascha Kurz
2021 arXiv   pre-print
We also derive general upper bounds for subcodes arising in those constructions.  ...  Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for A_q(n,d;k) are the topic of many recent research papers.  ...  Rank-metric codes of constant rank with a lower bound on the minimum rank-distance have been studied in [10] and generalized in [17, 33] .  ... 
arXiv:2109.07128v2 fatcat:6k2cfm7vjjgjvmuziqlmhrx57e

The interplay of different metrics for the construction of constant dimension codes

Sascha Kurz
2022 Advances in Mathematics of Communications  
We also derive general upper bounds for subcodes arising in those constructions.</p>  ...  Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage.  ...  Rank-metric codes of constant rank with a lower bound on the minimum rank-distance have been studied in [10] and generalized in [17, 33] .  ... 
doi:10.3934/amc.2021069 fatcat:bzk42a5qkbb5pazyppv6wjc3ay

Combining subspace codes [article]

Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese
2020 arXiv   pre-print
In the context of constant–dimension subspace codes, an important problem is to determine the largest possible size A_q(n, d; k) of codes whose codewords are k-subspaces of F_q^n with minimum subspace  ...  Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes.  ...  The work of the first, third and fourth author was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).  ... 
arXiv:1911.03387v2 fatcat:nxmocsaxcfhonanh5lt4qhkrye

Combining subspace codes

Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese
2021 Advances in Mathematics of Communications  
In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size A q (n, d; k) of codes whose codewords are k-subspaces of F n q with minimum subspace  ...  Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes.  ...  The work of the first, third and fourth author was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).  ... 
doi:10.3934/amc.2021007 fatcat:td7rm65vdfculdy3mzqxa3qsia

Projective Space Codes for the Injection Metric [article]

Azadeh Khaleghi, Frank R. Kschischang
2009 arXiv   pre-print
In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field.  ...  Using the code-construction framework of Etzion and Silberstein, new non-constant-dimension codes are constructed; these codes contain more codewords than comparable codes designed for the subspace metric  ...  In Theorem 3 we provide a lower bound on the dimension κ of the largest [S(v), κ, δ] Ferrers diagram rank-metric code obtained as a subcode of a linear MRD code. Theorem 3.  ... 
arXiv:0904.0813v2 fatcat:hv6cgrb4cvcefgazjnzbp63fhq

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.  ...  Sudhir Ghorparde for their valuable comments and suggestions on this work. I also thank the anonymous reviewer who introduced me to the notion of linear sets.  ... 
arXiv:1907.04372v3 fatcat:p4ilmaxnvfbahgxg7mwg6peyca

Bounds on Relative Generalized Hamming Weight

Zhuojun Zhuang, Keke Zhang, Zhen Jing, Bin Dai, Jia Huang
2020 IET Communications  
In the non-asymptotic setting, they present a new proof of the Griesmer bound on RGHW by residue code and introduce the new concept of relative chain condition.  ...  The rth relative generalised Hamming weight (RGHW) of an [n, k] linear code C and an [n, k 1 ] subcode C 1 , a generalisation of generalised Hamming weight (GHW), characterises code performances of wiretap  ...  In future work, we shall study more bounds on the first metric, such as asymptotic Hamming and Elias bounds, and construct good coding schemes with a large code length in all kinds of applications.  ... 
doi:10.1049/iet-com.2019.0903 fatcat:pniuabxxbrcxzc6pxnwhprgr54

Linear MSRD codes with Various Matrix Sizes and Unrestricted Lengths [article]

Hao Chen
2022 arXiv   pre-print
A sum-rank-metric code attaining the Singleton bound is called maximum sum-rank distance (MSRD). MSRD codes have been constructed for some parameter cases.  ...  In this paper we construct a linear MSRD code over an arbitrary field F_q with various matrix sizes n_1>n_2>⋯>n_t satisfying n_i ≥ n_i+1^2+⋯+n_t^2 for i=1, 2, ..., t-1 for any given minimum sum-rank distance  ...  This is an MRD code. We refer to [2, 11] for recent results on rank-metric codes and [13, 20] for recent results on MRD codes.  ... 
arXiv:2206.02330v3 fatcat:yeuam7cw55eszg2qzez763keqy

Cryptanalysis of a System Based on Twisted Reed-Solomon Codes [article]

Julien Lavauzelle, Julian Renner
2019 arXiv   pre-print
The algorithm is based on the recovery of the structure of subfield subcodes of twisted Reed--Solomon codes, and it always succeeds.  ...  In this paper, an efficient key-recovery attack is given on this variant of the McEliece cryptosystem.  ...  We would further like to thank the authors of the proposed cryptosystem [4] for validating our attack and pointing out a possible repair of the system with respect to our attack.  ... 
arXiv:1904.11785v1 fatcat:p7svrf6jvbeklofshvqnr4hjyy

Fast distributed video deduplication via locality-sensitive hashing with similarity ranking

Yeguang Li, Liang Hu, Ke Xia, Jie Luo
2019 EURASIP Journal on Image and Video Processing  
This paper proposes a flexible and fast distributed video deduplication framework based on hash codes.  ...  It is able to support the hash table indexing using any existing hashing algorithm in a distributed environment and can efficiently rank the candidate videos by exploring the similarities among the key  ...  Based on this similarity metric, we can easily rank all candidates in descending order.  ... 
doi:10.1186/s13640-019-0442-7 fatcat:d4vmavc22baqfjjaxdmqdpdeby

A New Rank Metric Codes Based Encryption Scheme [chapter]

Pierre Loidreau
2017 Lecture Notes in Computer Science  
The idea of the design mixes two existing approaches designing rank metric based encryption schemes.  ...  We design a new McEliece-like rank metric based encryption scheme from Gabidulin codes. We explain why it is not affected by the invariant subspace attacks also known as Overbeck's attacks.  ...  Concerning Goppa codes that are subfield subcodes of Generalized Reed-Solomon codes (GRS) on scrambles the structure by keeping the subcode formed with the binary vectors of the parent GRS code.  ... 
doi:10.1007/978-3-319-59879-6_1 fatcat:kqdhhgfszbfinpdfbdu24af6wy

Optimal Ferrers Diagram Rank-Metric Codes [article]

Antonia Wachter-Zeh, Tuvi Etzion
2014 arXiv   pre-print
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.  ...  COMBINING DIFFERENT FERRERS DIAGRAM RANK-METRIC CODES In this section, we show two possible ways to obtain new Ferrers diagram rank-metric codes based on rank-metric codes in subdiagrams.  ... 
arXiv:1405.1885v2 fatcat:jxtd2ndddjdmfa3rtq4ydtklzu
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