pith. sign in

arxiv: 2507.08755 · v2 · submitted 2025-07-11 · 💻 cs.IT · math.IT

Column Twisted Reed-Solomon Codes as MDS Codes

Wei Liu , Jinquan Luo , Puyin Wang , Dengxin Zhai This is my paper

Pith reviewed 2026-05-19 05:04 UTC · model grok-4.3

classification 💻 cs.IT math.IT MSC 94B05
keywords MDS codesReed-Solomon codestwisted Reed-Solomon codesSchur squarecode equivalencedual codesgenerator matrix
0
0 comments X

The pith

Column twisted Reed-Solomon codes formed by appending vectors to Reed-Solomon generators achieve the MDS property under stated conditions and reach lengths up to (q+3)/2.

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

This paper introduces column twisted Reed-Solomon codes constructed by appending column vectors to a generator matrix of a Reed-Solomon code. It establishes sufficient conditions under which these codes achieve the maximum distance separable property. The work shows that the Schur square of these codes has dimension 2k, which establishes they are not equivalent to Reed-Solomon codes. The construction supports code lengths up to (q+3)/2 for large odd prime power q, exceeding the (q+1)/2 limit of systematic twisted generalized Reed-Solomon codes. It also gives explicit descriptions of the dual codes.

Core claim

Column twisted Reed-Solomon codes, obtained by appending suitable column vectors to the generator matrix of a Reed-Solomon code, are MDS when the twisting parameters satisfy the given sufficient conditions. Their Schur square has dimension exactly 2k, so the codes are inequivalent to Reed-Solomon codes. The same construction yields MDS codes of length up to (q+3)/2 and supplies their duals.

What carries the argument

Column twisted Reed-Solomon codes formed by appending column vectors to a Reed-Solomon generator matrix.

Load-bearing premise

Specific twisting parameters and appended column vectors exist that satisfy the sufficient conditions for the MDS property at the claimed field sizes and lengths.

What would settle it

An explicit choice of odd prime power q together with length n equal to (q+3)/2 for which no choice of appended column vectors produces an MDS code would show the length claim fails.

read the original abstract

In this paper, we study column twisted Reed-Solomon(TRS) codes. We establish some sufficient conditions for these codes to be MDS and show that the dimension of their Schur square codes is $2k$. Consequently, these TRS codes are shown to be not equivalent to Reed-Solomon(RS) codes. Moreover, our construction offers more flexible parameters than existing twisted generalized Reed-Solomon(TGRS) code designs. For a large odd prime power $q$, systematically constructed TGRS codes are known to be limited to length $\frac{q+1}{2}$. By contrast, our column TRS construction supports code lengths up to $\frac{q+3}{2}$. Finally, we present the dual codes of column TRS codes. Overall, this work introduces a new method for constructing MDS codes by appending column vectors to some generator matrix of an RS code.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces column twisted Reed-Solomon (TRS) codes obtained by appending specific column vectors to a generator matrix of a Reed-Solomon code. It states sufficient conditions under which these codes are MDS, proves that the Schur square has dimension exactly 2k (implying the codes are inequivalent to Reed-Solomon codes), derives their duals, and claims that the construction supports lengths up to (q+3)/2 for large odd prime powers q, exceeding the (q+1)/2 limit of known systematically constructed twisted generalized Reed-Solomon codes.

Significance. If the sufficient conditions are rigorously verified and the existence of suitable twisting parameters is established for the extended lengths, the work would supply a new, more flexible explicit construction of MDS codes together with a clean algebraic distinction via the Schur-square dimension. The dual-code description adds completeness. The parameter extension beyond prior TGRS bounds is potentially useful for applications that require longer MDS codes over finite fields.

major comments (2)
  1. [Section 3 (MDS conditions and parameter existence)] The central claim that the construction yields MDS codes for lengths up to (q+3)/2 rests on the existence of twisting parameters (or appended column vectors) that make every k × k minor of the generator matrix nonzero. The manuscript asserts sufficient conditions on these parameters but does not supply an explicit choice or a determinant argument showing that the conditions remain satisfiable when the extra column is appended (in contrast to the TGRS case limited to (q+1)/2). This existence step is load-bearing for both the MDS property and the flexibility claim.
  2. [Section 4 (Schur-square dimension)] The proof that the Schur square has dimension 2k is used to conclude non-equivalence to Reed-Solomon codes. The argument should be checked to ensure it holds uniformly for all admissible twisting parameters and does not inadvertently reduce to a special case already covered by ordinary RS codes.
minor comments (2)
  1. [Section 2 (Construction)] Notation for the appended column vectors and the twisting parameters should be introduced once and used consistently; a small table summarizing the admissible ranges of q and n would improve readability.
  2. [Section 5 (Duals)] The dual-code derivation is stated without an accompanying example; adding a small explicit example for a modest q and k would help verify the formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the major comments point by point below and will revise the paper to incorporate clarifications and additional arguments where needed.

read point-by-point responses

  1. Referee: [Section 3 (MDS conditions and parameter existence)] The central claim that the construction yields MDS codes for lengths up to (q+3)/2 rests on the existence of twisting parameters (or appended column vectors) that make every k × k minor of the generator matrix nonzero. The manuscript asserts sufficient conditions on these parameters but does not supply an explicit choice or a determinant argument showing that the conditions remain satisfiable when the extra column is appended (in contrast to the TGRS case limited to (q+1)/2). This existence step is load-bearing for both the MDS property and the flexibility claim.

    Authors: We agree that an explicit choice of twisting parameters together with a determinant argument confirming satisfiability for the appended column would make the MDS claim for lengths up to (q+3)/2 more self-contained. In the revised version we will supply a concrete family of twisting vectors for sufficiently large odd prime-power q (built from a fixed nonzero element and a quadratic non-residue) and prove, via a Vandermonde-type expansion of the relevant minors, that none of the k × k minors vanish under the stated sufficient conditions. This will also clarify why the extra column can be added without violating the MDS property, thereby justifying the length extension beyond the (q+1)/2 bound of systematic TGRS constructions. revision: yes

  2. Referee: [Section 4 (Schur-square dimension)] The proof that the Schur square has dimension 2k is used to conclude non-equivalence to Reed-Solomon codes. The argument should be checked to ensure it holds uniformly for all admissible twisting parameters and does not inadvertently reduce to a special case already covered by ordinary RS codes.

    Authors: The current proof proceeds by exhibiting 2k linearly independent vectors in the Schur square that arise from the original RS generator rows together with the twisted columns; the argument uses only that the twisting parameters are nonzero and satisfy the MDS minor conditions. We will revise Section 4 to state explicitly that the linear-independence claim holds for every admissible (i.e., MDS-satisfying) choice of twisting parameters and to add a short remark showing that the Schur-square dimension collapses to k precisely when all twisting parameters are zero, which is excluded by our hypotheses. This guarantees the non-equivalence conclusion is uniform and does not collapse to the ordinary RS case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; MDS conditions derived independently via linear algebra

full rationale

The paper constructs column TRS codes by appending specific column vectors to an RS generator matrix and establishes sufficient conditions for the MDS property through explicit determinant non-vanishing requirements on the resulting k x k minors. These conditions are stated directly in terms of the twisting parameters and appended vectors without reducing to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The Schur square dimension result (equal to 2k) and consequent non-equivalence to RS codes are obtained from the same matrix construction. The length extension to (q+3)/2 is claimed under the sufficient conditions rather than by redefining the target property. No quoted step equates the output to the input by construction, and the derivation remains self-contained against external linear-algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard finite-field arithmetic and Reed-Solomon generator matrices; the new elements are the appended column vectors and twisting parameters whose existence is asserted under sufficient conditions not detailed here.

axioms (1)
  • standard math Reed-Solomon codes exist over finite fields of odd prime power order q and achieve the MDS property.
    Background fact from classical coding theory invoked to start the construction.

pith-pipeline@v0.9.0 · 5681 in / 1372 out tokens · 44976 ms · 2026-05-19T05:04:12.302415+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Two Families of Linear Codes Containing Non-GRS MDS Codes

    cs.IT 2025-12 unverdicted novelty 5.0

    Two new families of linear codes are built from modified GRS generator matrices, producing non-GRS MDS codes with derived parity-check matrices and self-duality conditions.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    Structural properties of twisted Reed-Solomon codes with applications to cryptography,

    P. Beelen, M. Bossert, S. Puchinger, and J. Rosenkilde, “Structural properties of twisted Reed-Solomon codes with applications to cryptography,”Proc. IEEE Int. Symp. Inform. Theory (ISIT), pp. 946-950, 2018

    work page 2018

  2. [2]

    On the covering radius of MDS codes,

    D. Bartoli, M. Giulietti, and I. Platoni, “On the covering radius of MDS codes,”IEEE Trans. Inf. Theory, vol. 61, no. 2, pp. 801–811, Feb. 2015

    work page 2015

  3. [3]

    Twisted Reed-Solomon codes,

    P. Beelen, S. Puchinger, and J. Rosenkilde n´ eNielsen, “Twisted Reed-Solomon codes,”Proc. IEEE Int. Symp. Inform. Theory (ISIT), pp. 336-340, 2017

    work page 2017

  4. [4]

    On parity-check matrices of twisted generalized Reed-Solomon codes,

    W. Cheng, “On parity-check matrices of twisted generalized Reed-Solomon codes,”IEEE Trans. Inform. Theory, vol. 70, no. 5, pp. 3213-3225, May 2024

    work page 2024

  5. [5]

    On codes, matroids, and secure multiparty computation from linear secret-sharing schemes,

    R. Cramer, V . Daza, I. Gracia, J. J. Urroz, G. Leander and J. Marti-Farre, “On codes, matroids, and secure multiparty computation from linear secret-sharing schemes,”IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2644–2657, Jun. 2008

    work page 2008

  6. [6]

    New constructions of MDS codes with asymptotically optimal repair,

    A. Chowdhury and A. Vardy, “New constructions of MDS codes with asymptotically optimal repair,” inProc. IEEE Int. Symp. Inf. Theory (ISIT), pp. 1944–1948, 2018

    work page 1944

  7. [7]

    A unified construction of finite geometries associated withq-clans in characteristic 2,

    W. E. Cherowitzo, C. M. O’Keefe, and T. Penttila, “A unified construction of finite geometries associated withq-clans in characteristic 2,”Adv. Geom., vol. 3, no. 1, pp. 1–21, 2003

    work page 2003

  8. [8]

    Euclidean and Hermitian LCD MDS codes,

    C. Carlet, S. Mesnager, C. Tang, and Y . Qi, “Euclidean and Hermitian LCD MDS codes,”Des., Codes Cryptogr., vol. 86, pp. 2605–2618, Nov. 2018. JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 2, DECEMBER 2023 11

    work page 2018

  9. [9]

    Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes,

    A. Couvreur, P. Gaborit, V . Gauthier-Uma˜ na, A. Otmani and J-P. Tillich, “Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes,”Des. Codes Cryptogr., vol. 73, pp. 641–666, 2014

    work page 2014

  10. [10]

    New self-dual codes from TGRS codes with generalltwists,

    Y . Ding and S. Zhu, “New self-dual codes from TGRS codes with generalltwists,”Adv. Math. Commun., vol. 19, no. 2, pp. 662-675, 2025

    work page 2025

  11. [11]

    The weights in MDS codes,

    M. F. Ezerman, M. Grassl, and P. Sol ´e, “The weights in MDS codes,”IEEE Trans. Inf. Theory, vol. 57, no. 1, pp. 392–396, Jan. 2011

    work page 2011

  12. [12]

    Deep holes of twisted Reed-Solomon codes,

    W. Fang and J. Xu, “Deep holes of twisted Reed-Solomon codes,”Proc. IEEE Int. Symp. Inform. Theory (ISIT), pp. 488-493, 2024

    work page 2024

  13. [13]

    A condition for ARCS and MDS codes,

    D. G. Glynn, “A condition for ARCS and MDS codes,”Des. Codes Cryptogr., vol. 58, no. 2, pp. 215–218, Feb. 2011

    work page 2011

  14. [14]

    Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes,

    G. Guo, R. Li, Y . Liu, and H. Song, “Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes,”Cryptogr. Commun., vol. 15, pp. 383-395, 2023

    work page 2023

  15. [15]

    New MDS or near-MDS self-dual codes,

    T. A. Gulliver, J. L. Kim, and Y . Lee, “New MDS or near-MDS self-dual codes,”IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4354–4360, Sept. 2008

    work page 2008

  16. [16]

    On twisted generalized Reed-Solomon codes withltwists,

    H. Gu and J. Zhang, “On twisted generalized Reed-Solomon codes withltwists,”IEEE Trans. Inform. Theory, vol. 70, no. 1, pp. 145-153, Jan. 2024

    work page 2024

  17. [17]

    MDS or NMDS LCD codes from twisted Reed-Solomon codes,

    D. Huang, Q. Yue, and Y . Niu, “MDS or NMDS LCD codes from twisted Reed-Solomon codes,”Cryptogr. Commun., vol. 15, pp. 221-237, 2023

    work page 2023

  18. [18]

    MDS or NMDS self-dual codes from twisted generalized Reed-Solomon codes,

    D. Huang, Q. Yue, Y . Niu, and X. Li, “MDS or NMDS self-dual codes from twisted generalized Reed-Solomon codes,”Designs, Codes Cryptogr., vol. 89, no. 9, pp. 2195-2209, 2021

    work page 2021

  19. [19]

    On the classification of MDS codes,

    J. I. Kokkala, D. S. Krotov, and R. J. Patric, “On the classification of MDS codes,”IEEE Trans. Inf. Theory, vol. 61, no. 12, pp. 6485–6492, Dec. 2015

    work page 2015

  20. [20]

    Construction of MDS self-dual codes over finite fields,

    K. Lebed, H. Liu, and J. Luo, “Construction of MDS self-dual codes over finite fields,”Finite Fields Appl., vol. 59, pp. 199–207, Sept. 2019

    work page 2019

  21. [21]

    Cryptanalysis of a system based on twisted Reed-Solomon codes,

    J. Lavauzelle and J. Renner, “Cryptanalysis of a system based on twisted Reed-Solomon codes,”Designs, Codes Cryptogr., vol. 88, no. 7, pp. 1285-1300, 2020

    work page 2020

  22. [22]

    Covering radii and deep holes of two classes of extended twisted GRS codes and their applications,

    Y . Li, S. Zhu, and Z. Sun, “Covering radii and deep holes of two classes of extended twisted GRS codes and their applications,”IEEE Trans. Inform. Theory, vol. 71, no. 5, pp. 3516-3530, May 2025

    work page 2025

  23. [23]

    Isomorphisms between Subiacoq-clan geometries,

    S. E. Payne, T. Penttila, and I. Pinneri, “Isomorphisms between Subiacoq-clan geometries,”Bull. Belg. Math. Soc. Simon Stevin, vol. 2, no. 2, pp. 197–222, 1995

    work page 1995

  24. [24]

    MDS multi-twisted Reed-Solomon codes with small dimensional hull,

    H. Singh and K. C. Meena, “MDS multi-twisted Reed-Solomon codes with small dimensional hull,”Cryptogr. Commun., vol. 16, pp. 557-578, 2024

    work page 2024

  25. [25]

    MDS, Near-MDS or 2-MDS self-dual codes via twisted generalized Reed-Solomon codes,

    J. Sui, Q. Yue, X. Li, and D. Huang, “MDS, Near-MDS or 2-MDS self-dual codes via twisted generalized Reed-Solomon codes,”IEEE Trans. Inform. Theory, vol. 68, no. 12, pp. 7832-7841, 2022

    work page 2022

  26. [26]

    Decoding algorithms of twisted GRS codes and twisted Goppa codes,

    H. Sun, Q. Yue, X. Jia, and C. Li, “Decoding algorithms of twisted GRS codes and twisted Goppa codes,”IEEE Trans. Inform. Theory, vol. 71, no. 2, pp. 1018-1027, 2025

    work page 2025

  27. [27]

    MDS and near-MDS codes via twisted Reed-Solomon codes,

    J. Sui, X. Zhu, and X. Shi, “MDS and near-MDS codes via twisted Reed-Solomon codes,”Designs, Codes Cryptogr., vol. 90, pp. 1937-1958, 2022

    work page 1937

  28. [28]

    Twisted Reed-Solomon codes with one-dimensional hull,

    Y . Wu, “Twisted Reed-Solomon codes with one-dimensional hull,”IEEE Commun. Lett., vol. 25, no. 2, pp. 383-386, 2021

    work page 2021

  29. [29]

    A class of twisted generalized Reed-Solomon codes,

    J. Zhang, Z. Zhou, and C. Tang, “A class of twisted generalized Reed-Solomon codes,”Designs, Codes Cryptogr., vol. 90, pp. 1649-1658, 2022

    work page 2022

  30. [30]

    Research on the construction of maximum distance separable codes via arbitrary twisted generalized Reed-Solomon codes,

    C. Zhao, W. Ma, T. Yan, and Y . Sun, “Research on the construction of maximum distance separable codes via arbitrary twisted generalized Reed-Solomon codes,”IEEE Trans. Inform. Theory, vol. 71, no. 2, pp. 1-1, 2025

    work page 2025

  31. [31]

    New MDS codes of non-GRS type and NMDS codes,

    Y . Zhi and S. Zhu, “New MDS codes of non-GRS type and NMDS codes,”Discrete Math., vol. 348, no. 5, 114436, 2025

    work page 2025

  32. [32]

    A class of double-twisted generalized Reed-Solomon codes,

    C. Zhu and Q. Liao, “A class of double-twisted generalized Reed-Solomon codes,”Finite Fields Appl., vol. 95, 102395, 2024

    work page 2024

  33. [33]

    The (+)-extended twisted generalized Reed-Solomon code,

    C. Zhu and Q. Liao, “The (+)-extended twisted generalized Reed-Solomon code,”Discrete Math., vol. 347, no. 2, 113749, 2024

    work page 2024