Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric
@article{Hrmann2021EfficientDO, title={Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric}, author={Felicitas H{\"o}rmann and Hannes Bartz}, journal={ArXiv}, year={2021}, volume={abs/2109.14943} }
. 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…
Figures from this paper
3 Citations
Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric
- Computer Science2022 IEEE International Symposium on Information Theory (ISIT)
- 2022
This work proposes the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding by incorporating erasures into the known syndrome-based Berlekamp-Massey-like decoder.
Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings
- Computer ScienceIFAC-PapersOnLine
- 2022
A fast divide-and-conquer variant of K¨otter–Nielsen–Høholdt (KNH) interpolation algorithm that inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gr¨obner basis of their kernel intersection and matches the previous best speeds for these tasks.
Covering Properties of Sum-Rank Metric Codes
- Computer Science2022 58th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
- 2022
This work intends to answer the question: what is the minimum cardinality of a code given a sum-rank covering radius, and shows the relations of this quantity between different metrics and provides several lower and upper bounds for sum-Rank metric codes.
References
SHOWING 1-10 OF 21 REFERENCES
Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants
- Computer ScienceArXiv
- 2022
Fast minimal approximant basis interpolation techniques are generalized to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length to be the first being able to correct errors beyond the unique decoding region in the sum-rank, sum-subspace and skew metric.
Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding
- Computer Science2021 IEEE International Symposium on Information Theory (ISIT)
- 2021
A heuristic argument and simulation results indicate that the list size is in fact one for most channel realizations up to the maximal decoding radius, and the best upper bound on the listsize is exponential.
Decoding of block and convolutional codes in rank metric
- Computer Science
- 2013
A special class of convolutional codes in rank metric is introduced and an efficient decoding algorithm for these codes is proposed, which is (partial) unit memory codes, built upon rank-metric block codes.
Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric
- Computer ScienceIEEE Transactions on Information Theory
- 2021
A skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials is described, which captures the bulk of the work in multiplication of skew polynomial rings and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity.
Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring
- Computer ScienceJournal of Algebra
- 2018
Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy
- Computer ScienceIEEE Transactions on Information Theory
- 2008
This paper presents an explicit construction of error-correcting codes of rate that can be list decoded in polynomial time up to a fraction of worst-case errors, and their methods directly yield better decoding algorithms for RS codes when errors occur in phased bursts.
List-decoding of subspace codes and rank-metric codes up to Singleton bound
- Computer Science2012 IEEE International Symposium on Information Theory Proceedings
- 2012
A folded version of Gabidulin codes analogous to the folded Reed-Solomon codes of Guruswami and Rudra is introduced along with a list-decoding algorithm for such codes that achieves the information theoretic bound on the decoding radius of a rank-metric code.
Efficient interpolation-based decoding of interleaved subspace and Gabidulin codes
- Computer Science2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)
- 2014
An interpolation-based decoding scheme for interleaved subspace codes is presented, an efficient interpolation procedure for the required linearized multivariate polynomials is presented and a computationally- and memory-efficient root-finding algorithm for the probabilistic unique decoder is proposed.
Residues of skew rational functions and linearized Goppa codes
- Computer Science, MathematicsArXiv
- 2019
The main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials) and prove a skew analogue of the residue formula and a skew analog of the classical formula of change of variables for residues.
Algebraic decoding of folded Gabidulin codes
- Computer ScienceDes. Codes Cryptogr.
- 2017
An efficient interpolation-based decoding algorithm that can correct rank errors beyond half the minimum rank distance for any code rate and an upper bound on the average list size of folded Gabidulin codes and on the decoding failure probability of the decoder are given.