A new rank metric for convolutional codes

Almeida, Paulo José Fernandes; Napp, Diego

doi:10.1007/s10623-020-00808-w

“…This survey has discussed block codes in the rank metric. There is also a line of work on convolutional codes in the rank metric (Wachter et al, 2011;Wachter-Zeh and Sidorenko, 2012;Wachter-Zeh et al, 2015;Napp et al, 2017b;Napp et al, 2017a;Napp et al, 2018;Almeida and Napp, 2021). These codes are considered in the related sum-rank metric, which can be seen as a mix of the Hamming and rank metric.…”

Section: Discussionmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz

¹

,

Holzbaur

²

,

Liu

³

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

“…This survey has discussed block codes in the rank metric. There is also a line of work on convolutional codes in the rank metric (Wachter et al, 2011;Wachter-Zeh and Sidorenko, 2012;Wachter-Zeh et al, 2015;Napp et al, 2017b;Napp et al, 2017a;Napp et al, 2018;Almeida and Napp, 2021). These codes are considered in the related sum-rank metric, which can be seen as a mix of the Hamming and rank metric.…”

Section: Discussionmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz¹,

Holzbaur²,

Liu³

et al. 2022

Preprint

0

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

“…Again, rank metric convolutional codes are typically defined over Fqm [D] (see [1,2,3,15,22]) as finitely generated Fqm [D]-submodules of Fqm [D] n described by…”

Section: A General Framework For Rank Metric Convolutional Codesmentioning

confidence: 99%

“…The work in [22] was pioneer in this direction by presenting the first class of unit-memory rank metric convolutional codes. Later, more properties and applications were investigated in [2,3,15], showing the potential of such a framework to spread redundancy across the codewords. However, there exist very few algebraic constructions of multi-shot network codes in comparison to the literature on one-shot network rank metric codes.…”

Section: Introductionmentioning

confidence: 99%

On the construction of MRD convolutional codes

Napp,

Pinto,

Santana

et al. 2024

Linear and Multilinear Algebra

Self Cite

0

View full text Add to dashboard Cite

The problem of building optimal block codes, such as MDS codes, over small fields has been an active area of research that led to several interesting conjectures. In the context of convolutional codes, optimal constructions, such as MDS or MDP, are very rare and all require very large finite fields. In this work, we focus on the problem of constructing optimal convolutional codes with respect to the rank distance, i.e., we study the construction of Maximum Rank Distance (MRD) convolutional codes. Considering convolutional codes within a very general framework, we present concrete novel classes of MRD convolutional codes for a large set of given parameters.

show abstract

A new rank metric for convolutional codes

Cited by 6 publications

References 32 publications

Rank-Metric Codes and Their Applications

Rank-Metric Codes and Their Applications

Rank-Metric Codes and Their Applications

On the construction of MRD convolutional codes

Contact Info

Product

Resources

About