On the Similarities Between Generalized Rank and Hamming Weights and Their Applications to Network Coding

Martínez-Peñas, Umberto

doi:10.1109/tit.2016.2570238

Cited by 46 publications

(84 citation statements)

References 37 publications

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“…11] are essentially equivalent. However, we adopt the approach in [35], which gives one restriction and one shortening for each support rather than each matrix, since different generator matrices of the same support give equivalent codes. In this way, the properties of the restricted and shortened codes are related to properties of the corresponding supports.…”

Section: Restriction Shortening Change Of Bases and Pre-shorteningmentioning

confidence: 99%

“…This result follows directly from [ Proof. With the definitions and results from Sections 2 and 3, the proof can be translated mutatis mutandis from those in [35,Th. 3] or [43,Prop.…”

Section: Generalized Sum-rank Weightsmentioning

confidence: 99%

“…We then extend the classical operations of restriction and shortening to linear sum-rank codes, establish their duality and give estimates on the sum-rank parameters of the codes that they provide. We give three applications: 1) We introduce generalized sum-rank weights, give their basic properties and applications, and establish a hierarchy of bounds on them, extending a previous connection between bounds on generalized Hamming and rank weights [35]; 2) We connect the MSRD property with sum-rank supports, and prove that duals, shortenings and restrictions of MSRD codes give again MSRD codes; 3) We introduce and characterize degenerateness and effective length of linear sum-rank codes, showing in particular that MSRD codes are never sum-rank degenerate.…”

Section: Introductionmentioning

confidence: 98%

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Theory of supports for linear codes endowed with the sum-rank metric

Martínez-Peñas

2019

Des. Codes Cryptogr.

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The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (support spaces) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: 1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; 2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; 3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.

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Section: Restriction Shortening Change Of Bases and Pre-shorteningmentioning

confidence: 99%

“…This result follows directly from [ Proof. With the definitions and results from Sections 2 and 3, the proof can be translated mutatis mutandis from those in [35,Th. 3] or [43,Prop.…”

Section: Generalized Sum-rank Weightsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Theory of supports for linear codes endowed with the sum-rank metric

Martínez-Peñas

2019

Des. Codes Cryptogr.

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“…The result is a variant of [6, Theorem 1] and was explicitly presented in [10, Theorem 3.2 and Corollary 3.3] using the Bruhat decomposition for matrices. It is also an immediate consequence of the more general results [20,Theorem 2] or [20,Theorem 6].…”

Section: Block Codesmentioning

confidence: 67%

Systematic maximum sum rank codes

Almeida

Martínez-Peñas

Napp

2020

Finite Fields and Their Applications

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In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.

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“…The perspective is how the original error is transferred into the propagated errors based on different distance metrics. Martínez-Peñas [25] compares Hamming metric and the rank metric in network coding. X is transmitted with network coding scheme.…”

Section: Existing Error-correcting Methods In Network Codingmentioning

confidence: 99%

The combination of sparse learning and list decoding of subspace codes for error correction in random network coding

Zhang

Cai

Zhang

et al. 2018

J Wireless Com Network

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The network coding can spread a single original error over the whole network. The simulation shows that the propagated error mostly all the time pollute just 100% of the received packets at the sink if Hamming distance is adopted. If subspace codes are adopted, usually the propagated error will not pollute 100% of the received packets in the sense of subspace distance. However, it also usually pollutes 90% of received packets which is a high error ratio. Even if the rank code and the subspace code are adopted, these existing schemes based on traditional block codes can correct corrupted errors no more than C/2 because of the limitation of the block coding where C is the max flow min cut. It is an agent to find a dense error correction method in random network coding. List decoding of subspace codes can correct C k -1 errors where k is the size of information. When C k is big, many errors in the sense of subspace distance can be corrected. However, the solution of list decoding is not unique. John Wright proposed a dense error correction technique based on L1 minimization, which can recover nearly 100% of the corrupted observations. In our proposal, the original packets are coded with John Wright's coding matrix, and then, the coded message is coded again with subspace codes. In the sink, the decoding procedures about list decoding of subspace codes and John Wright's scheme are performed. At last, the unique solution is achieved even though there are dense propagated errors in random network coding.

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On the Similarities Between Generalized Rank and Hamming Weights and Their Applications to Network Coding

Cited by 46 publications

References 37 publications

Theory of supports for linear codes endowed with the sum-rank metric

Theory of supports for linear codes endowed with the sum-rank metric

Systematic maximum sum rank codes

The combination of sparse learning and list decoding of subspace codes for error correction in random network coding

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