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Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes.

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Title: Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes.
Authors: Hörmann, Felicitas, Bartz, Hannes
Source: Designs, Codes & Cryptography; Mar2024, Vol. 92 Issue 3, p553-586, 34p
Subject Terms: REED-Solomon codes, DECODING algorithms, MONTE Carlo method, BLOCK codes, LINEAR network coding, COMPUTATIONAL complexity
Abstract: The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm. [ABSTRACT FROM AUTHOR]
Copyright of Designs, Codes & Cryptography is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes.
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  Data: Designs, Codes & Cryptography; Mar2024, Vol. 92 Issue 3, p553-586, 34p
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  Data: <searchLink fieldCode="DE" term="%22REED-Solomon+codes%22">REED-Solomon codes</searchLink><br /><searchLink fieldCode="DE" term="%22DECODING+algorithms%22">DECODING algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22MONTE+Carlo+method%22">MONTE Carlo method</searchLink><br /><searchLink fieldCode="DE" term="%22BLOCK+codes%22">BLOCK codes</searchLink><br /><searchLink fieldCode="DE" term="%22LINEAR+network+coding%22">LINEAR network coding</searchLink><br /><searchLink fieldCode="DE" term="%22COMPUTATIONAL+complexity%22">COMPUTATIONAL complexity</searchLink>
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  Data: The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
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  Data: <i>Copyright of Designs, Codes & Cryptography is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10623-023-01214-8
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        Text: English
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      – SubjectFull: REED-Solomon codes
        Type: general
      – SubjectFull: DECODING algorithms
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      – SubjectFull: MONTE Carlo method
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      – SubjectFull: BLOCK codes
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      – SubjectFull: LINEAR network coding
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      – SubjectFull: COMPUTATIONAL complexity
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              M: 03
              Text: Mar2024
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              Y: 2024
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