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Syndrome-Based Error-Erasure Decoding of Interleaved Linearized Reed-Solomon Codes

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Pith's one-line read Syndrome-based decoding of vertically and horizontally interleaved linearized Reed–Solomon codes can correct full errors, row erasures, and column erasures together, with guaranteed and probabilistic unique-decoding radii and quantified…

desk verdict Solid syndrome-based decoders for VILRS and HILRS, but the headline error-erasure guarantees rest on an explicitly deferred reduction, so treat those radius and failure claims as conditional. read the letter →

arxiv 2411.19101 v1 pith:SHYAKX44 submitted 2024-11-28 cs.IT math.IT

Felicitas Hörmann , Hannes Bartz This is my paper · ORCID
classification cs.ITmath.IT MSC 94B3594B2711T71
keywords linearizedReed–Solomoncodesinterleavedsum-rankmetricerror-erasuredecodingsyndrome-basedskewpolynomialsverticalinterleavinghorizontal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Vertically and horizontally interleaved linearized Reed–Solomon (LRS) codes can each be decoded by a syndrome-based algorithm that handles full errors, row erasures, and column erasures in one joint key equation instead of treating erasures as a separate pre-processing step. The paper claims guaranteed unique decoding whenever the number of full errors satisfies $t_{\mathcal{F}}\le \tfrac{1}{2}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$ (up to the exact per-component terms), and probabilistic unique decoding up to $t_{\mathcal{F}}\le \tfrac{s}{s+1}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$ with an explicit failure-probability bound. These are claimed to be the first syndrome-based and the first error-erasure decoders for both interleaved code families, and they unify the vertical and horizontal settings through a dual error-locator / error-span polynomial framework. If the claims hold, erasure side information translates directly into decoding power, and the interleaving gain $s/(s+1)$ known from rank-metric codes now covers joint error-erasure scenarios in the sum-rank metric.

What carries the argument

The machinery lives in the skew-polynomial ring $\mathbb{F}_{q^m}[x;\theta^{-1}]$ with generalized operator evaluation, and in two dual key equations. For VILRS codes the central object is the error-locator polynomial (ELP), the minimal skew polynomial vanishing on the error locators; the decoder sets up the ELP key equation $\lambda_{\mathcal{F}}\cdot s_{\mathcal{CR},j}\equiv \psi_j \pmod{x^{n-k}}$, where $s_{\mathcal{CR},j}$ is an auxiliary component-syndrome polynomial built from the received syndrome together with the known erasure information. For HILRS codes the central object is the error-span polynomial (ESP), with the dual key equation $\sigma_{\mathcal{F}}\cdot s_{\mathcal{RC},j}\equiv \omega_j \pmod{x^{n-k}}$; the decoder first recovers error values instead of error locations. Partial ELPs and ESPs encode the known column spaces of row erasures and the known row spaces of column erasures, and these partial polynomials are multiplied into the auxiliary syndromes. The key equations are solved by multisequence skew-feedback shift-register synthesis; roots of the recovered skew polynomial are found with a Skachek–Roth-like algorithm; and the remaining linear systems, which have generalized Moore matrices, are solved with a Gabidulin-like algorithm.

What would settle it

For parameters such as $q=3$, $m=4$, $\ell=2$, $n=(4,4)$, $k=3$, $s=4$, run the VILRS error-erasure decoder with $t_{\mathcal{F}}=4$ plus row and column erasures chosen so that $\tau^*_{\mathrm{vert}}=\tau_{\max}=4$; if the observed failure rate exceeds the predicted bound $\kappa_q^{\ell+1}q^{-m((s+1)(\tau_{\max}-\tau^*)+1)}$ by more than the factor of about 3–5 seen in the paper's error-only simulations, the unproven reduction in Theorem 4 does not hold.

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Extended reading notes

Core claim

The paper's central claim is that a single syndrome-based decoder, built from a Berlekamp–Massey-like key equation, recovers the transmitted codeword of a VILRS or HILRS code from an additive sum-rank error of weight $\tau=t_{\mathcal{F}}+t_{\mathcal{R}}+t_{\mathcal{C}}$, decomposed into full errors, row erasures, and column erasures. For vertical interleaving the key object is the error-locator polynomial (ELP), and the decoder first recovers error locations; for horizontal interleaving it is the error-span polynomial (ESP), and the decoder first recovers error values. Erasure knowledge is incorporated directly: known column spaces of row erasures produce partial component ESPs, and known row spaces of column erasures produce partial ELPs, which are multiplied into auxiliary component-syndrome polynomials in the key equation. The paper establishes guaranteed unique decoding for $t_{\mathcal{F}}\le \tfrac{1}{2}(n-k-\max_j t_{\mathcal{R},j}-t_{\mathcal{C}})$ in the vertical case and $t_{\mathcal{F}}\le \tfrac{1}{2}(n-k-t_{\mathcal{R}}-\max_j t_{\mathcal{C},j})$ in the horizontal case, and states probabilistic unique decoding for larger weights up to $t_{\mathcal{F}}\le \tfrac{s}{s+1}(n-k-\bar{t}_{\mathcal{R}}-t_{\mathcal{C}})$ (with the roles of $\mathcal{R}$ and $\mathcal{C}$ swapped for horizontal interleaving). The failure probability is bounded by $\kappa_q^{\ell+1} q^{-m((s+1)(\tau_{\max}-\tau^*)+1)}$, where $\tau^*$ is the effective error weight and $\tau_{\max}=\tfrac{s}{s+1}(n-k)$; Monte Carlo simulations for error-only decoding confirm the bound is tight within a constant factor.

Load-bearing premise

The probabilistic radii and failure bounds for the error-erasure decoders rest on the assertion, made in the proofs of Theorems 4 and 8, that the auxiliary syndrome polynomials can be rewritten as ordinary error-only syndromes; that rewriting is stated without proof and deferred to follow-up work.

Editorial extensions

If this is right

  • VILRS and HILRS codes become the first interleaved sum-rank code families with syndrome-based error-erasure decoding, extending the known error-only decoders.
  • The complexity bounds $O(sn^2)$ for error-only and $\widetilde{O}(sn^2)$ for error-erasure decoding mean the erasure-aware key equations cost no additional asymptotic order in most parameter regimes.
  • Each row or column erasure reduces the guaranteed full-error radius by one unit, so the decoding radius is $n-k$ minus the erasure budget, halved for full errors.
  • The probabilistic radius extends the decodable full-error weight by the interleaving factor $s/(s+1)$, matching the gain known from interleaved Gabidulin codes.
  • The unified ELP/ESP presentation exposes a systematic duality—shared row space for vertical interleaving, shared column space for horizontal—that should make future decoding results transferable between the two settings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A completed proof of the deferred reduction would close the gap in Theorems 4 and 8 and would give the same rigorous failure bound for non-interleaved LRS error-erasure decoding, where the bound is currently inherited from the rank-metric setting.
  • The joint treatment of erasures is directly relevant to code-based cryptography: side information about an error's row or column space is exactly what these decoders convert into a larger decoding radius, so security analyses of sum-rank cryptosystems should account for this erasure-aware capability.
  • A natural testable extension is heterogeneous interleaving, which the authors name as future work; the same ELP/ESP key-equation architecture appears flexible enough to allow different component codes per row or column without reworking the syndrome algebra.
  • Lifted versions of VILRS and HILRS codes could inherit the error-erasure capability for multishot network coding, where lost packets play the role of erasures and corrupted packets play the role of full errors.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies vertically and horizontally interleaved linearized Reed–Solomon (VILRS and HILRS) codes in the sum-rank metric. It presents syndrome-based error-only decoders for both families, with a guaranteed decoding radius of (n−k)/2 and a probabilistic radius of s(n−k)/(s+1), together with explicit failure-probability bounds. It then formulates error-erasure decoders that jointly handle full errors, row erasures, and column erasures, claiming a guaranteed radius t_F ≤ (n−k−t_R−t_C)/2 and a probabilistic radius t_F ≤ s/(s+1)(n−k−t_R−t_C), with the exact radius depending on the interleaving direction. The paper further gives average-case complexity estimates and Monte Carlo simulations for the error-only failure probability.

Significance. If the main claims are established, the paper provides a useful unified treatment of vertical and horizontal interleaving for LRS codes and extends syndrome-based decoding to mixed error-erasure channels. The error-only decoders are derived in detail: Theorem 1, Theorem 5, Lemma 2, and Lemma 4 contain full proofs of the key equations and of the rank-based failure-probability bounds, and the simulations in Section V give quantitative evidence for the tightness of those bounds. The advertised central novelty, however, is the error-erasure generalization, and that part currently rests on an explicitly deferred reduction. The paper would be a solid contribution once that missing argument is supplied.

major comments (3)
  1. [Section III-D, proof of Theorem 4]
  2. [Section IV-D, proof of Theorem 8]
  3. [Section V, Table I]
minor comments (4)
  1. [Algorithm 4, line 20]
  2. [Section V, Eq. (63)]
  3. [Sections III-D and IV-D, Eqs. (33) and (58)]
  4. [Abstract and Conclusion]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: key equations are derived in-paper, and the explicitly deferred error-erasure reduction is a proof gap, not a circular step.

full rationale

The claimed decoding results are not circular. The paper derives the ELP/ESP key equations (Theorems 1, 3, 5, 7) by direct coefficient computations from the syndrome expressions (15), (25), (43), and (51), and the decoding radii follow from counting equations against unknowns in the equivalent linear systems (18)/(32)/(45)/(57), culminating in (21), (33), (48), and (58). The guaranteed-radii parts of Lemmas 2 and 4 are proved in-paper by the rank decomposition of the syndrome matrix. The probabilistic failure bound imports [35, Lem. 6-7]; this is a self-citation (Bartz is a co-author), but it is an independent published result with stated assumptions that do not contain the present claim, and Section V's Monte Carlo runs reproduce the same bound in the error-only setting, so it functions as external evidence rather than a circular premise. The error-erasure Theorems 4 and 8 contain an explicitly deferred assertion that the auxiliary syndrome polynomials can be reinterpreted as modified error-only syndromes; the proofs state that the details will be presented in follow-up work. That is a missing proof/technical gap in the central probabilistic error-erasure claim, but it is not a circularity: the assertion is neither defined into existence nor obtained by renaming a fitted parameter, and the key equations themselves are proved unconditionally. No step in the derivation chain reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard structural assumptions of LRS codes and on the uniform error model from [35]. The main unproved dependence is the reduction of the error-erasure failure bound to the error-only case, which is a derivation gap rather than a free parameter or invented entity.

assumptions (4)
  • domain assumption The channel error is uniformly distributed over the set of matrices (or vectors) of a given sum-rank weight.
    Used in Lemma 2, Lemma 4, and the main theorems to derive failure probability bounds. This is an explicit modeling assumption, not a mathematical axiom.
  • domain assumption The field automorphism theta has fixed field exactly F_q, and evaluation parameters are chosen from distinct nontrivial conjugacy classes.
    Ensures generalized Moore matrices have full rank and minimal skew polynomials have the expected degree. Invoked throughout Sections II-IV, based on prior literature [1], [43], [45].
  • standard math The skew polynomial ring F_qm[x;theta] is a left and right Euclidean ring.
    Used for divisions, gcd/lclm, and the Berlekamp-Massey-like synthesis in Section VI. This is a standard result from Ore's theory.
  • standard math The parity-check matrix of an LRS code can be written as a generalized Moore matrix M_{theta^-1}^{n-k}(h)_{tilde xi}.
    Stated in Section II-C based on [5], [46]. Used in all syndrome computations and in the key equation derivations.

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Pith. "Pith review of Syndrome-Based Error-Erasure Decoding of Interleaved Linearized Reed-Solomon Codes." pith.science (2026). https://pith.science/paper/SHYAKX44

@misc{pith2026241119101,
  author       = {Pith},
  title        = {Pith review of: Syndrome-Based Error-Erasure Decoding of Interleaved Linearized Reed-Solomon Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHYAKX44}},
  note         = {Machine review of arXiv:2411.19101}
}
abstract

Linearized Reed--Solomon (LRS) codes are sum-rank-metric codes that generalize both Reed--Solomon and Gabidulin codes. We study vertically and horizontally interleaved LRS (VILRS and HILRS) codes whose codewords consist of a fixed number of stacked or concatenated codewords of a chosen LRS code. Our unified presentation of results for horizontal and vertical interleaving is novel and simplifies the recognition of resembling patterns. This paper's main results are syndrome-based decoders for both VILRS and HILRS codes. We first consider an error-only setting and then present more general error-erasure decoders, which can handle full errors, row erasures, and column erasures simultaneously. Here, an erasure means that parts of the row space or the column space of the error are already known before decoding. We incorporate this knowledge directly into Berlekamp--Massey-like key equations and thus decode all error types jointly. The presented error-only and error-erasure decoders have an average complexity in $O(sn^2)$ and $\widetilde{O}(sn^2)$ in most scenarios, where $s$ is the interleaving order and $n$ denotes the length of the component code. Errors of sum-rank weight $\tau=t_{\mathcal{F}}+t_{\mathcal{R}}+t_{\mathcal{C}}$ consist of $t_{\mathcal{F}}$ full errors, $t_{\mathcal{R}}$ row erasures, and $t_{\mathcal{C}}$ column erasures. Their successful decoding can be guaranteed for $t_{\mathcal{F}}\leq\tfrac{1}{2}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$, where $n$ and $k$ represent the length and the dimension of the component LRS code. Moreover, probabilistic decoding beyond the unique-decoding radius is possible with high probability when $t_{\mathcal{F}}\leq\tfrac{s}{s+1}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$ holds for interleaving order $s$. We give an upper bound on the failure probability for probabilistic unique decoding and showcase its tightness via Monte Carlo simulations.

Figures

Figures reproduced from arXiv: 2411.19101 by the authors.

Figure 1
Figure 1. Visualization of the sum-rank error decomposition i [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the sum-rank weight for vertically a [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Visualization of the observed decoding-failure pro [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

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Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [12]

    Error and Erasure C orrecting Algorithms for Rank Codes,

    E. M. Gabidulin and N. I. Pilipchuk, “Error and Erasure C orrecting Algorithms for Rank Codes,” Designs, Codes and Cryptography , vol. 49, no. 1–3, pp. 105–122, 2008

    work page 2008

  2. [19]

    Error-Erasur e Decoding of Linearized Reed–Solomon Codes in the Sum-Rank Metric,

    F. H¨ ormann, H. Bartz, and S. Puchinger, “Error-Erasur e Decoding of Linearized Reed–Solomon Codes in the Sum-Rank Metric,” in 2022 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2022, pp. 7–12

    work page 2022

  3. [1]

    Skew and Linearized Reed–Solomo n Codes and Maximum Sum Rank Distance Codes over any Division Ring,

    U. Mart´ ınez-Pe˜ nas, “Skew and Linearized Reed–Solomo n Codes and Maximum Sum Rank Distance Codes over any Division Ring,” Journal of Algebra , vol. 504, pp. 587–612, 2018

    work page 2018

  4. [2]

    A Unified Construction of Space- Time Codes with Optimal Rate-Diversity Tradeoff,

    H.-F. Lu and P . V . Kumar, “A Unified Construction of Space- Time Codes with Optimal Rate-Diversity Tradeoff,” IEEE Transactions on Information Theory, vol. 51, no. 5, pp. 1709–1730, 2005

    work page 2005

  5. [3]

    Multishot Codes f or Network Coding: Bounds and a Multilevel Construction,

    R. W. N´ obrega and B. F. Uchˆ oa-Filho, “Multishot Codes f or Network Coding: Bounds and a Multilevel Construction,” i n 2009 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2009, pp. 428–432

    work page 2009

  6. [4]

    Multishot Codes for Network Coding Using Rank-Metr ic Codes,

    ——, “Multishot Codes for Network Coding Using Rank-Metr ic Codes,” in 2010 Third IEEE International W orkshop on Wireless Network Coding. IEEE, 2010, pp. 1–6

    work page 2010

  7. [5]

    Reliable an d Secure Multishot Network Coding Using Linearized Reed–So lomon Codes,

    U. Mart´ ınez-Pe˜ nas and F. R. Kschischang, “Reliable an d Secure Multishot Network Coding Using Linearized Reed–So lomon Codes,” IEEE Transactions on Information Theory , vol. 65, no. 8, pp. 4785–4803, 2019

    work page 2019

  8. [6]

    Fast Decoding of Lifted Inter leaved Linearized Reed–Solomon Codes for Multishot Networ k Coding,

    H. Bartz and S. Puchinger, “Fast Decoding of Lifted Inter leaved Linearized Reed–Solomon Codes for Multishot Networ k Coding,” Designs, Codes and Cryptography, vol. 92, no. 8, pp. 2379–2421, 2024

    work page 2024

Show all 57 references
  1. [7]

    Private Information Retrieval f rom Locally Repairable Databases with Colluding Servers,

    U. Mart´ ınez-Pe˜ nas, “Private Information Retrieval f rom Locally Repairable Databases with Colluding Servers,” in 2019 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2019, pp. 1057–1061

    2019

  2. [8]

    Universal a nd Dynamic Locally Repairable Codes with Maximal Recoverab ility via Sum-Rank Codes,

    U. Mart´ ınez-Pe˜ nas and F. R. Kschischang, “Universal a nd Dynamic Locally Repairable Codes with Maximal Recoverab ility via Sum-Rank Codes,” IEEE Transactions on Information Theory , vol. 65, no. 12, pp. 7790–7805, 2019

    2019

  3. [9]

    Generic Dec oding in the Sum-Rank Metric,

    S. Puchinger, J. Renner, and J. Rosenkilde, “Generic Dec oding in the Sum-Rank Metric,” in 2020 IEEE International Symposium on Information Theory (ISIT). IEEE, 2020, pp. 54–59

    2020

  4. [10]

    Distingui shing and Recovering Generalized Linearized Reed–Solomon Codes,

    F. H¨ ormann, H. Bartz, and A.-L. Horlemann, “Distingui shing and Recovering Generalized Linearized Reed–Solomon Codes,” in Code-Based Cryptography: CBCrypto 2022 . Lecture Notes in Computer Science, Springer, 2023, pp. 1–2 0

    2022

  5. [11]

    Information-Set Decoding with Hin ts,

    A.-L. Horlemann, S. Puchinger, J. Renner, T. Schamberg er, and A. Wachter-Zeh, “Information-Set Decoding with Hin ts,” in Code-Based Cryptography: CBCrypto 2021 . Lecture Notes in Computer Science, Springer, 2022, pp. 60– 83

    2021

  6. [13]

    List and Unique Error-Erasu re Decoding of Interleaved Gabidulin Codes with Interpolat ion Techniques,

    A. Wachter-Zeh and A. Zeh, “List and Unique Error-Erasu re Decoding of Interleaved Gabidulin Codes with Interpolat ion Techniques,” Designs, Codes and Cryptography, vol. 73, no. 2, pp. 547–570, 2014

    2014

  7. [14]

    A Rank-Met ric Approach to Error Control in Random Network Coding,

    D. Silva, F. R. Kschischang, and R. K¨ otter, “A Rank-Met ric Approach to Error Control in Random Network Coding,” IEEE Transactions on Information Theory, vol. 54, no. 9, pp. 3951–3967, 2008

    2008

  8. [15]

    Error and Erasure Decoding of R ank-Codes with a Modified Berlekamp–Massey Algorithm,

    G. Richter and S. Plass, “Error and Erasure Decoding of R ank-Codes with a Modified Berlekamp–Massey Algorithm,” in 5th International ITG Conference on Source and Channel Coding (SCC) , 2004, pp. 203–210

    2004

  9. [16]

    RankSi gn: An Efficient Signature Algorithm Based on the Rank Metric ,

    P . Gaborit, O. Ruatta, J. Schrek, and G. Z´ emor, “RankSi gn: An Efficient Signature Algorithm Based on the Rank Metric ,” in Post-Quantum Cryptography: PQCrypto 2014 . Lecture Notes in Computer Science, Springer, 2014, pp. 88– 107

    2014

  10. [17]

    Using Rank-Metric Code s for Error Correction in Random Network Coding,

    D. Silva and F. R. Kschischang, “Using Rank-Metric Code s for Error Correction in Random Network Coding,” in 2007 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2007, pp. 796–800

    2007

  11. [18]

    Error Control for Network Coding,

    D. Silva, “Error Control for Network Coding,” Ph.D. dis sertation, University of Toronto, 2009

    2009

  12. [20]

    Maximum Sum-Rank Distance Codes over Finite Chain Rings,

    U. Mart´ ınez-Pe˜ nas and S. Puchinger, “Maximum Sum-Rank Distance Codes over Finite Chain Rings,” IEEE Transactions on Information Theory , vol. 70, no. 6, pp. 3878–3890, 2024

    2024

  13. [21]

    Randomize d Decoding of Linearized Reed–Solomon Codes Beyond the Uniq ue Decoding Radius,

    T. Jerkovits, H. Bartz, and A. Wachter-Zeh, “Randomize d Decoding of Linearized Reed–Solomon Codes Beyond the Uniq ue Decoding Radius,” in 2023 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2023, pp. 820–825

    2023

  14. [22]

    Support-Guessing Decoding Algorithms in the Sum- Rank Metric,

    ——, “Support-Guessing Decoding Algorithms in the Sum- Rank Metric,” arXiv preprint arXiv:2410.15806 , 2024

    2024 arXiv

  15. [23]

    Decoding Rank Errors Beyo nd the Error-Correcting Capability,

    P . Loidreau and R. Overbeck, “Decoding Rank Errors Beyo nd the Error-Correcting Capability,” in Tenth International W orkshop on Algebraic and Combinatorial Coding Theory (ACCT) , 2006

    2006

  16. [24]

    Decoding of Block and Convolutional C odes in Rank Metric,

    A. Wachter-Zeh, “Decoding of Block and Convolutional C odes in Rank Metric,” Ph.D. dissertation, Ulm University an d Universit´ e Rennes 1, 2013

    2013

  17. [25]

    Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric,

    H. Bartz, T. Jerkovits, S. Puchinger, and J. Rosenkilde , “Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric,” IEEE Transactions on Information Theory , vol. 67, no. 8, pp. 5026–5050, 2021

    2021

  18. [26]

    On Transform-Domain Error and Erasure Correction by Gabidulin Codes,

    W. Li, V . Sidorenko, and D. Silva, “On Transform-Domain Error and Erasure Correction by Gabidulin Codes,” Designs, Codes and Cryptography , vol. 73, no. 2, pp. 571–586, 2014

    2014

  19. [27]

    Decoding Interleaved Gab idulin Codes and Multisequence Linearized Shift-Register Synthesis,

    V . Sidorenko and M. Bossert, “Decoding Interleaved Gab idulin Codes and Multisequence Linearized Shift-Register Synthesis,” in 2010 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2010, pp. 1148–1152

    2010

  20. [28]

    Skew-Feedba ck Shift-Register Synthesis and Decoding Interleaved Gabi dulin Codes,

    V . R. Sidorenko, L. Jiang, and M. Bossert, “Skew-Feedba ck Shift-Register Synthesis and Decoding Interleaved Gabi dulin Codes,” IEEE Transactions on Information Theory , vol. 57, no. 2, pp. 621–632, 2011. 33

    2011

  21. [29]

    Row Reduction Applied to Decoding of Rank-Metric and Subspace Codes,

    S. Puchinger, J. Rosenkilde n´ e Nielsen, W. Li, and V . Si dorenko, “Row Reduction Applied to Decoding of Rank-Metric and Subspace Codes,” Designs, Codes and Cryptography , vol. 82, no. 1–2, pp. 389–409, 2016

    2016

  22. [30]

    Decoding Interleaved Gabidul in Codes using Alekhnovich’s Algorithm,

    S. Puchinger, S. M¨ uelich, D. M¨ odinger, J. Rosenkilde n´ e Nielsen, and M. Bossert, “Decoding Interleaved Gabidul in Codes using Alekhnovich’s Algorithm,” Electronic Notes in Discrete Mathematics , vol. 57, pp. 175–180, 2017

    2017

  23. [31]

    Durandal: A Rank Metric Based Signature Scheme,

    N. Aragon, O. Blazy, P . Gaborit, A. Hauteville, and G. Z´ emor, “Durandal: A Rank Metric Based Signature Scheme,” in Advances in Cryptology: Eurocrypt 2019. Lecture Notes in Computer Science, Springer, 2019, pp. 728 –758

    2019

  24. [32]

    LIGA: A Cr yptosystem Based on the Hardness of Rank-Metric List and Int erleaved Decoding,

    J. Renner, S. Puchinger, and A. Wachter-Zeh, “LIGA: A Cr yptosystem Based on the Hardness of Rank-Metric List and Int erleaved Decoding,” Designs, Codes and Cryptography , vol. 89, no. 6, pp. 1279–1319, 2021

    2021

  25. [33]

    LowMS: A New Rank Metric Code-Based K EM without Ideal Structure,

    N. Aragon, V . Dyseryn, P . Gaborit, P . Loidreau, J. Renner, and A. Wachter-Zeh, “LowMS: A New Rank Metric Code-Based K EM without Ideal Structure,” Designs, Codes and Cryptography , vol. 92, no. 4, pp. 1075–1093, 2023

    2023

  26. [34]

    Interlea ving Loidreau’s Rank-Metric Cryptosystem,

    J. Renner, S. Puchinger, and A. Wachter-Zeh, “Interlea ving Loidreau’s Rank-Metric Cryptosystem,” in 2019 XVI International Symposium on Problems of Redundancy in Information and Control Systems (REDUNDAN CY), 2019, pp. 127–132

    2019

  27. [36]

    Decoding of Interleaved Linearized Reed–Solomon Codes with Applications to Network Coding,

    ——, “Decoding of Interleaved Linearized Reed–Solomon Codes with Applications to Network Coding,” in 2021 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2021, pp. 160–165

    2021

  28. [37]

    On Decoding Hi gh-Order Interleaved Sum-Rank-Metric Codes,

    T. Jerkovits, F. H¨ ormann, and H. Bartz, “On Decoding Hi gh-Order Interleaved Sum-Rank-Metric Codes,” in Code-Based Cryptography: CBCrypto 2022 . Lecture Notes in Computer Science, Springer, 2023, pp. 90–1 09

    2022

  29. [38]

    An Error-Code Perspective on Metzner–Kapturowsk i-like Decoders,

    ——, “An Error-Code Perspective on Metzner–Kapturowsk i-like Decoders,” arXiv preprint arXiv:2201.01339 , 2024

    2024 arXiv

  30. [39]

    Fast Gao-Like Decoding of Hor izontally Interleaved Linearized Reed–Solomon Codes,

    F. H¨ ormann and H. Bartz, “Fast Gao-Like Decoding of Hor izontally Interleaved Linearized Reed–Solomon Codes,” in Code-Based Cryptography: CBCrypto 2023 . Lecture Notes in Computer Science, Springer, 2023, pp. 14– 34

    2023

  31. [40]

    Elliptic Periods for Finite Fields,

    J.-M. Couveignes and R. Lercier, “Elliptic Periods for Finite Fields,” Finite Fields and Their Applications , vol. 15, no. 1, pp. 1–22, 2009

    2009

  32. [41]

    On a Special Class of Polynomials,

    O. Ore, “On a Special Class of Polynomials,” Transactions of the American Mathematical Society , vol. 35, no. 3, pp. 559–584, 1933

    1933

  33. [42]

    Theory of Non-Commutative Polynomials,

    ——, “Theory of Non-Commutative Polynomials,” Annals of Mathematics , pp. 480–508, 1933

    1933

  34. [43]

    V andermonde and Wronskian Matr ices over Division Rings,

    T.-Y . Lam and A. Leroy, “V andermonde and Wronskian Matr ices over Division Rings,” Journal of Algebra , vol. 119, no. 2, pp. 308–336, 1988

    1988

  35. [44]

    Pseudo Linear Transformations and Evaluati on in Ore Extensions,

    A. Leroy, “Pseudo Linear Transformations and Evaluati on in Ore Extensions,” Bulletin of the Belgian Mathematical Society - Simon Stevin , vol. 2, no. 3, pp. 321–347, 1995

    1995

  36. [45]

    Residues of Skew Rational Functions and Lin earized Goppa Codes,

    X. Caruso, “Residues of Skew Rational Functions and Lin earized Goppa Codes,” arXiv preprint arXiv:1908.08430v1 , 2019

    1908 arXiv

  37. [46]

    Duals of Linearized Reed–Solo mon Codes,

    X. Caruso and A. Durand, “Duals of Linearized Reed–Solo mon Codes,” Designs, Codes and Cryptography , vol. 91, no. 1, pp. 241–271, 2022

    2022

  38. [47]

    Generic De coding in the Sum-Rank Metric,

    S. Puchinger, J. Renner, and J. Rosenkilde, “Generic De coding in the Sum-Rank Metric,” IEEE Transactions on Information Theory , vol. 68, no. 8, pp. 5075–5097, 2022

    2022

  39. [48]

    Equalities and Inequal ities for Ranks of Matrices,

    G. Matsaglia and G. P . H. Styan, “Equalities and Inequal ities for Ranks of Matrices,” Linear and Multilinear Algebra , vol. 2, no. 3, pp. 269–292, 1974

    1974

  40. [49]

    Efficient Decodi ng of Interleaved Low-Rank Parity-Check Codes,

    J. Renner, T. Jerkovits, and H. Bartz, “Efficient Decodi ng of Interleaved Low-Rank Parity-Check Codes,” in 2019 XVI International Symposium on Problems of Redundancy in Information and Control Systems ( REDUNDANCY), 2019, pp. 121–126

    2019

  41. [50]

    The Sage Developers, SageMath, the Sage Mathematics Software System (Version 10 .4), 2024, https://www.sagemath.org

    2024

  42. [51]

    Fast Skew-Feedback Shift -Register Synthesis,

    V . Sidorenko and M. Bossert, “Fast Skew-Feedback Shift -Register Synthesis,” Designs, Codes and Cryptography , vol. 70, no. 1–2, pp. 55–67, 2012

    2012

  43. [52]

    E. R. Berlekamp, Algebraic Coding Theory (Revised Edition) . World Scientific, 2015

    2015

  44. [53]

    V erfahren und Decoder zur Feh lerkorrektur einer empfangenen Nachricht,

    F. H¨ ormann and H. Bartz, “V erfahren und Decoder zur Feh lerkorrektur einer empfangenen Nachricht,” German Patent DE102 022 204 213B3, 2023, available at https://depatisnet.dpma.de/DepatisNet/de patisnet?action=bibdat&docid=DE102022204213B3

    2023

  45. [54]

    Probabilistic Algorithm for Finding Roots of Linearized Polynomials,

    V . Skachek and R. M. Roth, “Probabilistic Algorithm for Finding Roots of Linearized Polynomials,” Designs, Codes and Cryptography , vol. 46, no. 1, pp. 17–23, 2008

    2008

  46. [55]

    Theory of Codes with Maximum Rank Dist ance,

    E. M. Gabidulin, “Theory of Codes with Maximum Rank Dist ance,” Problems of Information Transmission , vol. 21, no. 1, pp. 1–12, 1985

    1985

  47. [56]

    Complexity of Decoding Gabidu lin Codes,

    M. Gadouleau and Z. Y an, “Complexity of Decoding Gabidu lin Codes,” in 2008 42nd Annual Conference on Information Sciences and Sys tems. IEEE, 2008, pp. 1081–1085

    2008

  48. [57]

    Improved Syndrom e Decoding of Interleaved Subspace Codes,

    H. Bartz, M. Meier, and V . Sidorenko, “Improved Syndrom e Decoding of Interleaved Subspace Codes,” in 11th International ITG Conference on Systems, Communications and Coding (SCC) , 2017

    2017

  49. [58]

    Improved Syndrome Decoding of Lifted L-interleaved Gabidulin Codes,

    H. Bartz and V . Sidorenko, “Improved Syndrome Decoding of Lifted L-interleaved Gabidulin Codes,” Designs, Codes and Cryptography , vol. 87, no. 2, pp. 547–567, 2019

    2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.