Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

Martínez-Peñas, Umberto; Kschischang, Frank R.

doi:10.1109/allerton.2018.8635644

Cited by 37 publications

(104 citation statements)

References 36 publications

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“…The results in Figure 5 show, that the upper bound in (93) gives a good estimate of the actual decoding failure probability of the decoder. For the same parameters a (non-interleaved) linearized Reed-Solomon code [7] (i.e. s = 1) can only correct errors of sum-rank weight up to t = 2 (BMD radius).…”

Section: E Comparison To Previous Work and Simulation Resultsmentioning

confidence: 99%

“…The ILRS codes from Definition 5 include LRS codes (denoted by LRS[β, a, ℓ; n, k]) as a special case [7]. Besides that, ILRS codes generalize several code families in the Hamming, rank and sum-rank metric.…”

Section: B Interleaved Linearized Reed-solomon Codesmentioning

confidence: 99%

“…In [3], [7] a sum-rank metric representation of the multishot operator channel in the spirit of [43], [70] was considered. This equivalent channel representation is more suitable for considering sum-rank metric based decoding of LILRS codes.…”

Section: A Multishot Network Codingmentioning

confidence: 99%

“…Note, that there exists other metrics for multishot network coding like e.g. the sum-injection distance (see [7,Definition 7]) which are related to the sum-subspace distance and are not considered in this work. Similar to [42] we define the (γ, δ) reachability for the multishot operator channel.…”

Section: A Multishot Network Codingmentioning

confidence: 99%

“…Linearized Reed-Solomon codes have recently shown to provide reliable and secure coding schemes for multi-shot network coding [7]. Furthermore, there is a construction [8] of locally repairable codes with maximal recoverability (also known as partial MDS codes) based on linearized Reed-Solomon codes, which attains the smallest known field size among all existing code constructions for a wide range of code parameters.…”

Section: Introductionmentioning

confidence: 99%

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Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

Bartz¹,

Puchinger²

2022

Preprint

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We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants and propose efficient decoding schemes that can correct errors beyond the unique decoding radius in the sum-rank, sum-subspace and skew metric. The proposed interpolationbased scheme for ILRS codes can be used as a list decoder or as a probabilistic unique decoder that corrects errors of sum-rank up to t ≤ s s+1 (n − k), where s is the interleaving order, n the length and k the dimension of the code. Upper bounds on the list size and the decoding failure probability are given where the latter is based on a novel Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew metric.We generalize fast minimal approximant basis interpolation techniques to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length.Up to our knowledge, the presented decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-rank, sum-subspace and skew metric. The results for the proposed decoding schemes are validated via Monte Carlo simulations.

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Section: E Comparison To Previous Work and Simulation Resultsmentioning

confidence: 99%

Section: B Interleaved Linearized Reed-solomon Codesmentioning

confidence: 99%

Section: A Multishot Network Codingmentioning

confidence: 99%

Section: A Multishot Network Codingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

Bartz¹,

Puchinger²

2022

Preprint

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Pseudo-Random Error-Correcting Codes in Network Coding

Vladimirov

2023

EAI/Springer Innovations in Communication and Computing

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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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Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

Cited by 37 publications

References 36 publications

Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

Pseudo-Random Error-Correcting Codes in Network Coding

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

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