A decoding algorithm for twisted Gabidulin codes

Rosenthal, Joachim; Randrianarisoa, Tovohery Hajatiana

doi:10.1109/isit.2017.8007034

Cited by 11 publications

(6 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This equality together with those in (13) also gives a unique solution λ 1 ,¨¨¨, λ t . This case was recently discussed in [26], [20] for u " 1 and in [27] for u ą 1; ‚ when Γ " tpa, 0q, p0, bq : Norm q n {q paq P I, Norm q n {q pbq P GF pqqzp´1q nk I, a, b P GF pq n qu for some subset I Ď GF pqq, this corresponds to the nonadditive partition code given in [16]. We shall intensively investigate this case in the sequel.…”

Section: B Interpolation-based Decodingmentioning

confidence: 88%

Interpolation-based Decoding of Nonlinear Maximum Rank Distance Codes

2019

2019 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

show abstract

Section: B Interpolation-based Decodingmentioning

confidence: 88%

Interpolation-based Decoding of Nonlinear Maximum Rank Distance Codes

2019

2019 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

show abstract

“…Decoding of Sheekey's twisted Gabidulin codes and its additive variants was studied in (Randrianarisoa and Rosenthal, 2017;Randrianarisoa, 2017;Li, 2019;Li and Kadir, 2019;Kadir et al, 2021;Kadir and Li, 2020). The general decoding principle of (Randrianarisoa and Rosenthal, 2017;Randrianarisoa, 2017) is to solve a linear system of equations with one fewer equation than usual (cf. Section 2.8).…”

Section: Further Classes Of Mrd Codesmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz

Holzbaur

Liu

et al. 2022

FNT in Communications and Information Theory

View full text Add to dashboard Cite

The rank metric measures the distance between two matrices by the rank of their difference. 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. 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. This survey gives a general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.Codes in the rank metric have drawn increasing interest in recent years due to their application to cryptography, distributed storage and network coding. The survey by Gorla and Ravagnani (2018) mainly focuses on combinatorial properties of rank-metric codes, while the one by Sheekey (2019a) considers MRD codes and their properties. In particular, Gorla and Ravagnani (2018) and Gorla ( 2019) treat (amongst others): isometries, anticodes, duality, MacWilliams identities, generalized weights in the rank metric. We will therefore not considers these topics in this survey. An English version of the textbook by Gabidulin (2021) has been published recently which contains a collection of Gabidulin's results in the area of rank-metric codes. Our survey deals shortly with properties of rank-metric codes, but the main focus is on their decoding and their applications to code-based cryptography, storage, and network coding.In this chapter, we give an introduction to rank-metric codes, their properties and decoding. In Section 2.1, we provide the basic notation used in this survey. Section 2.2 defines linearized polynomials and recalls some basic properties. In Section 2.3, we define the rank metric and state bounds on the cardinality of rank-metric codes, i.e., spherepacking, Gilbert-Varshamov, and Singleton-like bounds. Section 2.4, 2.5 and 2.6 provide the weight distribution of rank metric codes, constant-rank codes and covering property of rank-metric codes, respectively. Section 2.7 defines Gabidulin codes, proves their minimum rank distance and gives generator and parity-check matrices. In Section 2.8, we describe syndrome-based decoding of Gabidulin codes, i.e., we prove the key equation, show how to solve it and how to reconstruct the final error. We also outline error-erasure decoding and summarize known fast decoders. In Section 2.9, we discuss results on list decoding of Gabidulin codes. Further, we introduce interleaved Gabidulin codes (Section 2.10), folded Gabidulin codes (Section 2.11) and summarize further classes of MRD codes (Section 2.13).

show abstract

“…When the rank of the error vector reaches the maximum unique decoding radius, syndrome-based decoding approach works only for MRD codes that are linear over the main extension field. Randrianarisoa in [24,26] , gave an interpolation based decoding algorithm for twisted Gabidulin codes. Later this idea was adopted to decode additive generalized twisted Gabidulin codes and Trombetti-Zhou rank metric codes [10,11].…”

Section: Introductionmentioning

confidence: 99%

Encoding and Decoding of Several Optimal Rank Metric Codes

Kadir¹,

Li²,

Zullo³

2022

Preprint

View full text Add to dashboard Cite

This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to ⌊(d − 1)/2⌋ in polynomial-time, where d is the minimum distance of the code.

show abstract

A decoding algorithm for twisted Gabidulin codes

Cited by 11 publications

References 5 publications

Interpolation-based Decoding of Nonlinear Maximum Rank Distance Codes

Interpolation-based Decoding of Nonlinear Maximum Rank Distance Codes

Rank-Metric Codes and Their Applications

Encoding and Decoding of Several Optimal Rank Metric Codes

Contact Info

Product

Resources

About