Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric

Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality. We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding. The presented syndrome-based Berlekamp-Massey-like error-erasure decoder can correct $t_F$ full errors, $t_R$ row erasures and $t_C$ column erasures up to $2t_F + t_R + t_C \leq n-k$ in the sum-rank metric requiring at most $\mathcal{O}(n^2)$ operations in $\mathbb{F}_{q^m}$, where $n$ is the code's length and $k$ its dimension. We show how the proposed decoder can be used to correct errors in the sum-subspace metric that occur in (noncoherent) multishot network coding.Comment: 6 pages, presented at ISIT 202

Maximum Sum-Rank Distance Codes over Finite Chain Rings

In this work, maximum sum-rank distance (MSRD) codes and linearized Reed-Solomon codes are extended to finite chain rings. It is proven that linearized Reed-Solomon codes are MSRD over finite chain rings, extending the known result for finite fields. For the proof, several results on the roots of skew polynomials are extended to finite chain rings. These include the existence and uniqueness of minimum-degree annihilator skew polynomials and Lagrange interpolator skew polynomials. A general cubic-complexity sum-rank Welch-Berlekamp decoder and a quadratic-complexity sum-rank syndrome decoder (under some assumptions) are then provided over finite chain rings. The latter also constitutes the first known syndrome decoder for linearized Reed--Solomon codes over finite fields. Finally, applications in Space-Time Coding with multiple fading blocks and physical-layer multishot Network Coding are discussed

Systematic maximum sum rank codes

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.publishe

Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

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 interpolation-based 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 $t\leq\frac{s}{s+1}(n-k)$, 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.Comment: submitted to IEEE Transactions on Information Theory, 57 pages, 10 figure

Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric

Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography. The sum-rank analogs of Reed-Solomon and Gabidulin codes are linearized Reed-Solomon codes. We show how to construct h-folded linearized Reed-Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius. The presented decoder can be used for either list or probabilistic unique decoding and requires at most O(sn^2) operations in F_{q^m}, where s<=h is an interpolation parameter and n denotes the length of the unfolded code. We derive a heuristic upper bound on the failure probability of the probabilistic unique decoder and verify the results via Monte Carlo simulations

Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric

Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography. The sum-rank analogs of Reed-Solomon and Gabidulin codes are linearized Reed-Solomon codes. We show how to construct h-folded linearized Reed-Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius. The presented decoder can be used for either list or probabilistic unique decoding and requires at most O(sn^2) operations in F_{q^m}, where s<=h is an interpolation parameter and n denotes the length of the unfolded code. We derive a heuristic upper bound on the failure probability of the probabilistic unique decoder and verify the results via Monte Carlo simulations