pith. sign in

arxiv: 2606.25662 · v1 · pith:BJPGILBFnew · submitted 2026-06-24 · 💻 cs.IT · math.IT

The MDS or NMDS for Modified GRS codes with flexible hull dimensions and lengths

Zhonghao Liang , Qunying Liao , Jun Zhang , Xiaoping Li This is my paper

Pith reviewed 2026-06-25 19:44 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords MGRS codesMDS codesNMDS codesHermitian hullEuclidean hullLCD codesweight distributionReed-Solomon codes
0
0 comments X

The pith

MGRS codes exist with adjustable Hermitian hull dimensions and lengths while being MDS or NMDS.

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

The paper studies modified generalized Reed-Solomon codes and their extensions. It proves that two families of these codes are either maximum distance separable or near maximum distance separable, gives exact conditions for the near case, and computes their weight distributions. It then builds four families that are Euclidean LCD codes or have one-dimensional Euclidean hulls. The central result is a constructive proof that MGRS codes can be made with any chosen Hermitian hull dimension and length within the allowed range.

Core claim

Modified generalized Reed-Solomon codes and their extensions can be constructed so that they are MDS or NMDS, achieve prescribed Euclidean hull dimensions of zero or one, and realize any desired Hermitian hull dimension together with flexible lengths.

What carries the argument

Modified generalized Reed-Solomon (MGRS) codes, built from a specific algebraic generator matrix that permits direct control of hull dimension via choice of evaluation points and multipliers.

If this is right

  • The weight distributions of the NMDS MGRS codes give exact formulas for the number of codewords of each weight.
  • Four explicit constructions yield Euclidean LCD MGRS codes or MGRS codes with Euclidean hull dimension exactly one.
  • The same codes are shown to be linearly inequivalent to elliptic-curve NMDS codes via the Schur product.
  • Examples confirm that the constructions work over concrete finite fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The ability to vary hull dimension independently of length may allow systematic search for codes with prescribed dual-containing properties.
  • Because the codes are non-GRS, they supply new candidates for applications that require distance and hull control simultaneously.
  • The inequivalence result suggests that MGRS codes enlarge the known pool of NMDS codes beyond those arising from elliptic curves.

Load-bearing premise

The algebraic form of the MGRS codes and the existence of finite fields and evaluation points that meet the required divisibility or nonzero conditions.

What would settle it

An explicit choice of field, length, and evaluation points satisfying the paper's divisibility conditions for which the Hermitian hull dimension is not equal to any of the predicted flexible values.

read the original abstract

Non-generalized Reed-Solomon (in short, non-GRS) type maximum distance separable (in short, MDS), near MDS (in short, NMDS), and linear complementary dual (in short, LCD) codes, as well as the hull of linear codes have interesting practical applications in cryptography and coding theory. In this paper, we focus on a class of non-GRS codes and its extended codes, i.e., modified generalized Reed-Solomon (MGRS) codes and extended MGRS (EMGRS) codes introduced by Wang et al. in 2026. Firstly, we prove that two classes of MGRS codes and EMGRS codes are either MDS or NMDS, derive the necessary and sufficient conditions for these codes to be NMDS, and then completely determine the weight distributions for one class of these NMDS MGRS or NMDS EMGRS codes. Secondly, we construct four classes of MGRS codes which are either Euclidean LCD codes or one-dimensional Euclidean hull codes. Thirdly, we constructively prove that there exist MGRS codes with flexible Hermitian hull dimensions and lengths. In addition, we illustrate the linearly inequivalence of NMDS MGRS codes and elliptic curve NMDS codes by Schur product. Finally, some corresponding examples are given.

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 studies modified generalized Reed-Solomon (MGRS) codes and extended MGRS (EMGRS) codes. It proves that two classes are MDS or NMDS, derives necessary and sufficient conditions for NMDS, determines weight distributions for one class of the NMDS codes, constructs four classes that are Euclidean LCD or one-dimensional Euclidean hull codes, constructively proves existence of MGRS codes with flexible Hermitian hull dimensions and lengths, illustrates linear inequivalence to elliptic curve NMDS codes via Schur product, and provides examples.

Significance. If the central existence claims hold without hidden restrictions on parameters, the work supplies new explicit families of non-GRS MDS/NMDS codes together with controlled Euclidean and Hermitian hull dimensions; the complete weight distributions and the constructive hull results would be useful for cryptographic applications that rely on hull properties. The paper supplies concrete constructions and examples rather than only existence arguments.

major comments (2)
  1. [third main result / Hermitian hull constructions] The third main result (constructive proof of MGRS codes with flexible Hermitian hull dimensions and lengths): the argument proceeds by selecting evaluation points that satisfy certain non-zero and divisibility conditions derived from the MGRS definition; however, the manuscript does not exhibit an explicit choice of field, length n, and target hull dimension that simultaneously meets all conditions for arbitrary prescribed hull dimensions, leaving open whether the achievable (n, hull-dimension) pairs are dense or subject to arithmetic obstructions.
  2. [first main result / weight distributions] The NMDS weight-distribution claim (first main result): the derivation of the weight enumerator for the identified class of NMDS MGRS/EMGRS codes is stated to be complete, yet the proof sketch invokes the same divisibility conditions used later for the hull constructions; if those conditions are not freely satisfiable, the weight-distribution formulas apply only to a restricted parameter set, weakening the utility of the enumeration.
minor comments (2)
  1. [abstract / introduction] The reference to Wang et al. (2026) appears in the abstract and introduction; the full citation should be supplied and its date verified.
  2. [preliminaries] Notation for the modified GRS generator matrix and the precise definition of the extension to EMGRS codes is introduced without a displayed equation; adding an explicit matrix form would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address the two major comments point by point below, providing clarifications on the explicitness of the constructions and the scope of the results. We propose targeted revisions to improve clarity without altering the core claims.

read point-by-point responses

  1. Referee: [third main result / Hermitian hull constructions] The third main result (constructive proof of MGRS codes with flexible Hermitian hull dimensions and lengths): the argument proceeds by selecting evaluation points that satisfy certain non-zero and divisibility conditions derived from the MGRS definition; however, the manuscript does not exhibit an explicit choice of field, length n, and target hull dimension that simultaneously meets all conditions for arbitrary prescribed hull dimensions, leaving open whether the achievable (n, hull-dimension) pairs are dense or subject to arithmetic obstructions.

    Authors: The proof supplies an explicit construction: for a prescribed hull dimension, the evaluation points are chosen from the multiplicative group of the finite field to satisfy the listed non-zero and divisibility conditions that arise directly from the definition of the modified GRS code. These conditions are arithmetic and can be satisfied over finite fields of sufficiently large order by standard counting arguments on cyclotomic cosets. While the manuscript does not list a single closed-form formula that works for every conceivable (q,n,k) triple, the method is constructive once parameters obeying the conditions are fixed. To address the concern about explicit choices and density, we will add a short subsection illustrating an algorithmic selection of the field and points for given target hull dimension, together with a concrete numerical example. revision: yes

  2. Referee: [first main result / weight distributions] The NMDS weight-distribution claim (first main result): the derivation of the weight enumerator for the identified class of NMDS MGRS/EMGRS codes is stated to be complete, yet the proof sketch invokes the same divisibility conditions used later for the hull constructions; if those conditions are not freely satisfiable, the weight-distribution formulas apply only to a restricted parameter set, weakening the utility of the enumeration.

    Authors: The weight-enumerator formulas are derived precisely for the subclass of NMDS codes that satisfy the necessary and sufficient divisibility conditions established earlier in the paper. The derivation is complete within this well-defined class; the formulas are not claimed to hold outside it. The overlap with the later hull-construction conditions is natural, as both results concern the same family of modified GRS codes. The utility of the enumeration is therefore tied to the same parameter regime in which the codes exist and are NMDS. We will insert a clarifying remark after the weight-distribution theorem that explicitly states the parameter restrictions and notes that the formulas apply exactly when the divisibility conditions hold. revision: partial

Circularity Check

0 steps flagged

No significant circularity; new constructions independent of cited definition

full rationale

The paper cites Wang et al. 2026 solely for the algebraic definition of MGRS/EMGRS codes and then derives MDS/NMDS conditions, weight distributions, LCD constructions, and existence of flexible Hermitian hull dimensions via explicit algebraic conditions and examples on that form. No self-citations from overlapping authors are load-bearing, no parameters are fitted and renamed as predictions, and no terms are defined circularly in terms of the claimed results. The central existence proofs are constructive and rely on external divisibility/non-zero conditions rather than reducing to the inputs by definition or self-citation chains. The derivation is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the algebraic definition of MGRS codes supplied by the 2026 Wang et al. reference and on standard facts about finite fields and duals of evaluation codes; no new free parameters or invented entities are introduced in the abstract itself.

axioms (1)
  • domain assumption The MGRS codes are defined exactly as in Wang et al. 2026 and the underlying field is finite with the usual arithmetic.
    All proofs and constructions presuppose this definition and the standard properties of finite fields.

pith-pipeline@v0.9.1-grok · 5763 in / 1276 out tokens · 25600 ms · 2026-06-25T19:44:26.020975+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

57 extracted references · 8 linked inside Pith

  1. [1]

    Some constructions of non-generalized Reed-Solomon MDS codes[J]

    Abdukhalikov, K., Ding, C., Verma, G. Some constructions of non-generalized Reed-Solomon MDS codes[J]. Discrete Mathematics, 2026: 115202

    2026

  2. [2]

    Near-MDS codes from elliptic curves[J]

    Aguglia, A., Giuzzi, L., Sonnino, A. Near-MDS codes from elliptic curves[J]. Designs, Codes and Cryptography, 2021, 89(5): 965-972

    2021

  3. [3]

    A multisecret-sharing scheme based on LCD codes[J]

    Alahmadi, A., Altassan, A., AlKenani, A., C ¸ alkavur, S., Shoaib, H., Sol ´e, P. A multisecret-sharing scheme based on LCD codes[J]. Mathematics, 2020, 8(2): 272

    2020

  4. [4]

    Complete (k, 3)-arcs from quartic curves[J]

    Bartoli, D., Giulietti, M., Zini, G. Complete (k, 3)-arcs from quartic curves[J]. Designs, Codes and Cryptography, 2016, 79: 487-505

    2016

  5. [5]

    K., Singh, H., Sarma, R

    Bhagat, A. K., Singh, H., Sarma, R. Row-Column Twisted Reed-Solomon codes[R]. arXiv preprint arXiv:2509.06919, 2025

    arXiv 2025

  6. [6]

    Beelen, P., Puchinger, S., Nielsen, J. R. Twisted Reed-Solomon codes[C]//2017 IEEE International Symposium on Information Theory (ISIT). IEEE, 2017: 336-340

    2017

  7. [7]

    Linearity and complements in projective space[J]

    Braun, M., Etzion, T., Vardy, A. Linearity and complements in projective space[J]. Linear Algebra and its Applications, 2013, 430(1): 57-70

    2013

  8. [8]

    Euclidean and Hermitian LCD MDS codes[J]

    Carlet C, Mesnager S, Tang C, et al. Euclidean and Hermitian LCD MDS codes[J]. Designs, Codes and Cryptography, 2018, 86(11): 2605-2618

    2018

  9. [9]

    On the hull-variation problem of equivalent linear codes[J]

    Chen, H. On the hull-variation problem of equivalent linear codes[J]. IEEE Transactions on Information Theory, 2023, 69(5): 2911-2922

    2023

  10. [10]

    Many non-Reed-Solomon type MDS codes from arbitrary genus algebraic curves[J]

    Chen H. Many non-Reed-Solomon type MDS codes from arbitrary genus algebraic curves[J]. IEEE Transactions on Information Theory, 2023, 70(7): 4856-4864

    2023

  11. [11]

    The twice-extended TGRS codes[J]

    Cheng, H., Zhu, S. The twice-extended TGRS codes[J]. Advances in Mathematics of Communications, 2026, 24: 35-60

    2026

  12. [12]

    Infinite families of near MDS codes holdingt-designs[J]

    Ding, C., Tang, C. Infinite families of near MDS codes holdingt-designs[J]. IEEE Transactions on Information Theory, 2020, 66(9): 5419-5428

    2020

  13. [13]

    An infinite family of linear codes supporting 4-designs[J]

    Tang, C., Ding, C. An infinite family of linear codes supporting 4-designs[J]. IEEE Transactions on Information Theory, 2021, 67(1): 244-254

    2021

  14. [14]

    MDS and NMDS codes from extended codes of extended twisted Reed-Solomon codes[J]

    Ma, J., Wang, J., Ding, Y . MDS and NMDS codes from extended codes of extended twisted Reed-Solomon codes[J]. Cryptography and Communications, 2026: 1-12

    2026

  15. [15]

    Four new families of NMDS codes with dimension 4 and their applications[J]

    Ding, Y ., Li, Y ., Zhu, S. Four new families of NMDS codes with dimension 4 and their applications[J]. Finite Fields and Their Applications, 2024, 99: 102495

    2024

  16. [16]

    M., Landjev, I

    Dodunekov, S. M., Landjev, I. N. Near-MDS codes over some small fields[J]. Discrete Mathematics, 2000, 213(1-3): 55-65

    2000

  17. [17]

    New classes of NMDS codes with dimension 3[J]

    Fan, C., Wang, A., Xu, L. New classes of NMDS codes with dimension 3[J]. Designs, Codes and Cryptography, 2024, 92: 397-418

    2024

  18. [18]

    Q., Song, C

    Feng, R. Q., Song, C. W. Combinatorics[M]. Beijing: Peking University Press, 2015

    2015

  19. [19]

    Relative hulls and their variations of narrow-sense primitive BCH codes[J]

    Gan, C., Li, C., Mesnager, S. Relative hulls and their variations of narrow-sense primitive BCH codes[J]. IEEE Transactions on Information Theory, 2026

    2026

  20. [20]

    The dimensions of Schur squares of HRS codes[R]

    Gu, H., Zhu, Z., Zhang, J. The dimensions of Schur squares of HRS codes[R]. arXiv preprint arXiv:2604.17864, 2026

    Pith/arXiv arXiv 2026

  21. [21]

    Guenda, K., Jitman, S., Gulliver, T. A. Constructions of good entanglement-assisted quantum error correcting codes[J]. Designs, Codes and Cryptography, 2018, 86(1): 121-136

    2018

  22. [22]

    Roth–Lempel NMDS codes of non-elliptic-curve type[J]

    Han, D., Fan, C. Roth–Lempel NMDS codes of non-elliptic-curve type[J]. IEEE Transactions on Information Theory, 2023, 69(9): 5670-5675

    2023

  23. [23]

    Explicit constructions of NMDS self-dual codes[J]

    Han, D., Zhang, H. Explicit constructions of NMDS self-dual codes[J]. Designs, Codes and Cryptography, 2024, 92(11): 3573-3585

    2024

  24. [24]

    On LCD codes and lattices[C]//2016 IEEE International Symposium on Information Theory (ISIT)

    Hou, X., Oggier, F. On LCD codes and lattices[C]//2016 IEEE International Symposium on Information Theory (ISIT). 2016: 1501-1505. 29

    2016

  25. [25]

    C., Pless, V

    Huffman, W. C., Pless, V . Fundamentals of Error-Correcting Codes[M]. Cambridge: Cambridge University Press, 2003

    2003

  26. [26]

    On(L,P)-Twisted Generalized Reed-Solomon Codes[J]

    Hu, Z., Wang, L., Li, N. On(L,P)-Twisted Generalized Reed-Solomon Codes[J]. IEEE Transactions on Information Theory, 2025

    2025

  27. [27]

    New families of non-Reed–Solomon MDS codes[J]

    Jin, L., Ma, L., Xing, C. New families of non-Reed–Solomon MDS codes[J]. IEEE Transactions on Information Theory, 2025, 72(2): 985-993

    2025

  28. [28]

    The extended codes of NMDS codes[J]

    Ma, Q., Kai, X., Zhu, S. The extended codes of NMDS codes[J]. Finite Fields and Their Applications, 2026, 114: 102844

    2026

  29. [29]

    Non-Reed-Solomon type cyclic MDS codes[J]

    Li, F., Chen, Y ., Chen, H. Non-Reed-Solomon type cyclic MDS codes[J]. IEEE Transactions on Information Theory, 2025

    2025

  30. [30]

    Counting subset sums of finite abelian groups[J]

    Li, J., Wan, D. Counting subset sums of finite abelian groups[J]. Journal of Combinatorial Theory, Series A, 2012, 119(1): 170-182

    2012

  31. [31]

    A family of linear codes that are either non-GRS MDS codes or NMDS codes[J]

    Li, Y ., Sun, Z., Zhu, S. A family of linear codes that are either non-GRS MDS codes or NMDS codes[J]. IEEE Transactions on Communications, 2025

    2025

  32. [32]

    Covering radii and deep holes of two classes of extended twisted GRS codes and their applications[J]

    Li, Y ., Zhu, S., Sun, Z. Covering radii and deep holes of two classes of extended twisted GRS codes and their applications[J]. IEEE Transactions on Information Theory, 2025

    2025

  33. [33]

    Two classes of NMDS codes from Roth-Lempel codes[J]

    Liang, Z., Liao, Q. Two classes of NMDS codes from Roth-Lempel codes[J]. Finite Fields and Their Applications, 2026, 111: 102779

    2026

  34. [34]

    The extended code for a class of generalized Roth-Lempel codes and their properties[J]

    Liang, Z., Liao, Q. The extended code for a class of generalized Roth-Lempel codes and their properties[J]. Discrete Mathematics, 2026, 67(8): 115084

    2026

  35. [35]

    The equivalent condition for GRL codes to be MDS, AMDS or self-dual[R]

    Liang, Z., Liao, Q. The equivalent condition for GRL codes to be MDS, AMDS or self-dual[R]. arXiv preprint arXiv:2506.03874, 2025

    arXiv 2025

  36. [36]

    Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs[R]

    Liang, Z., Huang, D., Liao, Q. Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs[R]. arXiv preprint arXiv:2603.16187, 2026

    arXiv 2026

  37. [37]

    Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix[R]

    Liu, S., Liu, H., Chen, B. Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix[R]. arXiv preprint arXiv:2507.20559, 2025

    arXiv 2025

  38. [38]

    Column twisted Reed-Solomon codes as MDS codes[R]

    Liu, W., Luo, J., Wang, P. Column twisted Reed-Solomon codes as MDS codes[R]. arXiv preprint arXiv:2507.08755, 2025

    Pith/arXiv arXiv 2025

  39. [39]

    Luo, G., Sok, L., Ezerman, M. F. On linear codes whose Hermitian hulls are MDS[J]. IEEE Transactions on Information Theory, 2024, 70(7): 4889-4904

    2024

  40. [40]

    On the equivalence of NMDS codes[J]

    Lu, J., Zhou, Y . On the equivalence of NMDS codes[J]. IEEE Transactions on Information Theory, 2025, 71(12): 9506-9515

    2025

  41. [41]

    Non-Reed-Solomon Type MDS Codes from Elliptic Curves[R]

    Wang, P., Liu, W., Luo, J. Non-Reed-Solomon Type MDS Codes from Elliptic Curves[R]. arXiv preprint arXiv:2509.04247, 2025

    Pith/arXiv arXiv 2025

  42. [42]

    Macdonald, I. G. Symmetric functions and Hall polynomials[M]. Oxford: Oxford University Press, 1998

    1998

  43. [43]

    The theory of error correcting codes

    MacWilliams, Florence Jessie, and Neil James Alexander Sloane. The theory of error correcting codes. V ol. 16. Elsevier, 1977

    1977

  44. [44]

    M., Lempel, A

    Roth, R. M., Lempel, A. A construction of non-Reed-Solomon type MDS codes[J]. IEEE Transactions on Information Theory, 1989, 35(3): 655-657

    1989

  45. [45]

    Finding the permutation between equivalent linear codes: The support splitting algorithm[J]

    Sendrier, N. Finding the permutation between equivalent linear codes: The support splitting algorithm[J]. IEEE Transactions on Information Theory, 2002, 46(4): 1193-1203

    2002

  46. [46]

    Recursive Structure of Hulls of PRM Codes[R]

    Song, Y ., Yue, Q. Recursive Structure of Hulls of PRM Codes[R]. arXiv preprint arXiv:2604.21808, 2026

    Pith/arXiv arXiv 2026

  47. [47]

    Shortest Embeddings of Linear Codes with Arbitrary Hull Dimension[R]

    Wang, J., Luo, J. Shortest Embeddings of Linear Codes with Arbitrary Hull Dimension[R]. arXiv preprint arXiv:2604.08843, 2026

    Pith/arXiv arXiv 2026

  48. [48]

    New Constructions of Non-GRS MDS Codes, Recovery and Determination Algorithms for GRS Codes[J]

    Wang, G., Liu, H., Luo, J. New Constructions of Non-GRS MDS Codes, Recovery and Determination Algorithms for GRS Codes[J]. IEEE Transactions on Information Theory, 2026

    2026

  49. [49]

    MDS and NMDS Codes from the Extended Twisted Generalized Reed-Solomon Codes[R]

    Wang, Y ., Chen, Y ., Yan, T. MDS and NMDS Codes from the Extended Twisted Generalized Reed-Solomon Codes[R]. arXiv preprint arXiv:2605.23329, 2026

    Pith/arXiv arXiv 2026

  50. [50]

    Constructions ofℓ-MDS Self-Dual Codes With Flexibleℓvia Deleted Generalized Reed–Solomon Codes[J]

    Wan, R., Zhu, S. Constructions ofℓ-MDS Self-Dual Codes With Flexibleℓvia Deleted Generalized Reed–Solomon Codes[J]. IEEE Transactions on Information Theory, 2025, 72(2): 961-984

    2025

  51. [51]

    More MDS codes of non-Reed-Solomon type[R]

    Wu, Y ., Heng, Z., Li, C. More MDS codes of non-Reed-Solomon type[R]. arXiv preprint arXiv:2401.03391, 2024

    arXiv 2024

  52. [52]

    Analysis of Roth–Lempel Codes[J]

    Xu, H., Zhou, H. Analysis of Roth–Lempel Codes[J]. IEEE Transactions on Information Theory, 2025, 72(1): 246-252

    2025

  53. [53]

    Non-GRS type MDS and AMDS codes from extended TGRS codes[R]

    Zhang, M., Yang, S., Zheng, Y . Non-GRS type MDS and AMDS codes from extended TGRS codes[R]. arXiv preprint arXiv:2604.05682, 2026

    Pith/arXiv arXiv 2026

  54. [54]

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

    Zhao, C., Ma, W., Yan, T. Research on the construction of maximum distance separable codes via arbitrary twisted generalized Reed-Solomon codes[J]. IEEE Transactions on Information Theory, 2025, 71(7): 5130-5143

    2025

  55. [55]

    Generalized Roth–Lempel Codes: NMDS Characterization, Hermitian Self-Orthogonality, and Quantum Constructions[J]

    Liu Q, Wu X, Zhou H. Generalized Roth–Lempel Codes: NMDS Characterization, Hermitian Self-Orthogonality, and Quantum Constructions[J]. arXiv preprint arXiv:2604.11350, 2026

    Pith/arXiv arXiv 2026

  56. [56]

    A secret sharing scheme based on near-MDS codes[C]//Proceedings of 30 the IC-NIDC

    Zhou, Y ., Wang, F., Xin, Y ., Qing, S., Yang, Y . A secret sharing scheme based on near-MDS codes[C]//Proceedings of 30 the IC-NIDC. 2009: 833-836

    2009

  57. [57]

    New MDS codes of non-GRS type and NMDS codes[J]

    Zhi, Y ., Zhu, S. New MDS codes of non-GRS type and NMDS codes[J]. Discrete Mathematics, 2025, 348(5): 114436

    2025