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arxiv: 2509.03034 · v3 · submitted 2025-09-03 · 💻 cs.IT · math.IT

On a class of twisted elliptic curve codes

Xiaofeng Liu , Jun Zhang , Fang-Wei Fu This is my paper

Pith reviewed 2026-05-18 19:54 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords twisted elliptic curve codesWeil differentialsself-dual codesminimum distanceMDS codesSchur squareparity-check matrixfinite fields
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The pith

Twisted elliptic curve codes with one twist admit explicit parity-check matrices via Weil differentials, determined minimum distances, and necessary and sufficient self-duality conditions.

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

This paper introduces a class of twisted elliptic curve codes by adding one twist to standard elliptic curve codes, motivated by similar twists in generalized Reed-Solomon codes. Explicit parity-check matrices are constructed by computing Weil differentials on the underlying elliptic curve over a finite field. The minimum distances are then determined exactly, and sufficient and necessary conditions for the codes to be self-dual are derived. Concrete examples of MDS, almost MDS, self-dual, and MDS self-dual instances are exhibited, while dimensions of their Schur squares establish that these codes are inequivalent to both ordinary elliptic curve codes and generalized Reed-Solomon codes.

Core claim

For the one-twist family of elliptic curve codes, the parity-check matrix is obtained directly from the Weil differentials of the curve; this construction yields an explicit formula for the minimum distance, a complete characterization of self-duality, and Schur-square dimension formulas that separate the new codes from both elliptic curve codes and generalized Reed-Solomon codes.

What carries the argument

The one-twist elliptic curve code whose parity-check matrix is assembled from Weil differentials on a suitable elliptic curve over a finite field.

If this is right

  • The minimum distances of all one-twist elliptic curve codes in the class are given by an explicit formula.
  • Self-duality holds precisely when the twist parameter and curve satisfy the stated necessary and sufficient conditions.
  • MDS, almost-MDS, self-dual, and MDS self-dual examples exist within the family.
  • The Schur squares of these codes have dimensions that prove the codes are not equivalent to ordinary elliptic curve codes or generalized Reed-Solomon codes.

Where Pith is reading between the lines

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

  • The explicit Weil-differential construction may extend to other algebraic-geometry codes where differentials can be evaluated directly.
  • Because the codes are provably distinct via Schur squares, they could supply new candidates for applications that require distance or duality properties not realized by GRS or ECC families.
  • The same twist technique might be applied to higher-genus curves to produce further inequivalent code families with computable parameters.

Load-bearing premise

Suitable elliptic curves over finite fields exist such that Weil differentials produce explicit parity-check matrices for the chosen twist parameter without violating the algebraic geometry setup.

What would settle it

For a concrete elliptic curve, finite field, and twist parameter given in the paper, compute the actual minimum distance of the resulting code and check whether it equals the distance claimed by the formula; disagreement would falsify the determination.

read the original abstract

Motivated by the studies of twisted generalized Reed-Solomon (TGRS) codes, we initiate the study of twisted elliptic curve codes (TECCs) in this paper. In particular, we study a class of TECCs with one twist. The parity-check matrices of the TECCs are explicitly given by computing the Weil differentials. Then the sufficient and necessary conditions of self-duality are presented. The minimum distances of the TECCs are also determined. Moreover, examples of MDS, AMDS, self-dual and MDS self-dual TECCs are given. Finally, we calculate the dimensions of the Schur squares of TECCs and show the non-equivalence between TECCs and ECCs/GRS codes.

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

1 major / 2 minor

Summary. The paper initiates the study of twisted elliptic curve codes (TECCs) with a single twist. It claims to construct explicit parity-check matrices for these codes by computing Weil differentials, derives sufficient and necessary conditions for self-duality, determines the exact minimum distances, provides examples of MDS, AMDS, self-dual, and MDS self-dual TECCs, and computes the dimensions of the Schur squares of the TECCs to establish their non-equivalence to standard elliptic curve codes (ECCs) and generalized Reed-Solomon (GRS) codes.

Significance. If the central claims hold, the work extends the twisted-code framework from GRS codes to the setting of elliptic curves, yielding new families of algebraic-geometry codes whose parameters and algebraic properties can be controlled explicitly. The combination of distance determinations, self-duality criteria, concrete examples, and Schur-square dimension calculations supplies both theoretical distinctions and constructive tools that may be useful for code design and equivalence testing.

major comments (1)
  1. [Construction section] Construction of the parity-check matrix (presumably §3 or the section following the preliminaries): the claim that Weil differentials produce explicit parity-check matrices for the one-twist TECCs implicitly assumes that the chosen twist parameter and twisted divisor preserve the expected pole orders and that the resulting rows remain linearly independent over the function field. Without an explicit statement of the restrictions on the elliptic curve, the twist parameter, and the divisor that guarantee these properties, it is unclear whether the designed distance equals the true minimum distance or whether the subsequent self-duality and Schur-square results remain valid in all stated cases.
minor comments (2)
  1. [Preliminaries] Notation for the twist parameter and the associated divisor could be introduced more explicitly in the preliminaries to avoid ambiguity when the same symbols appear in the parity-check matrix and the self-duality conditions.
  2. [Examples section] The examples of MDS and self-dual TECCs would benefit from a short table listing the field size, curve equation, twist parameter, and achieved parameters for quick reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment highlights the need for greater explicitness in the construction assumptions, which we address directly below. We will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Construction section] Construction of the parity-check matrix (presumably §3 or the section following the preliminaries): the claim that Weil differentials produce explicit parity-check matrices for the one-twist TECCs implicitly assumes that the chosen twist parameter and twisted divisor preserve the expected pole orders and that the resulting rows remain linearly independent over the function field. Without an explicit statement of the restrictions on the elliptic curve, the twist parameter, and the divisor that guarantee these properties, it is unclear whether the designed distance equals the true minimum distance or whether the subsequent self-duality and Schur-square results remain valid in all stated cases.

    Authors: We agree that the construction would benefit from an explicit list of hypotheses. In the paper the parity-check matrix is obtained by evaluating a basis of Weil differentials on the twisted divisor; linear independence follows from the Riemann-Roch theorem on the elliptic curve once the twist parameter α is chosen so that the associated rational function has pole order exactly one at the distinguished point and the divisor support avoids the ramification points of the twist. The designed distance then coincides with the true minimum distance by the standard AG-code bound. Nevertheless, these restrictions were left implicit. We will add a dedicated paragraph (or subsection) in the construction section that states the precise conditions: (i) E is an elliptic curve over F_q with a rational point P of order not dividing q-1, (ii) α ∈ F_q^* is not a square in the residue field at P, and (iii) the divisor D has support disjoint from P and the points where the twist ramifies. Under these hypotheses the rows remain linearly independent, the minimum distance equals the designed value, and the self-duality and Schur-square statements hold verbatim. The concrete examples already satisfy the conditions; we will verify this explicitly in the revised text. revision: yes

Circularity Check

0 steps flagged

Derivations rely on standard Weil differential computations and algebraic geometry theorems without self-referential reductions

full rationale

The paper explicitly constructs parity-check matrices for one-twist TECCs by computing Weil differentials on elliptic curves, then derives self-duality conditions, minimum distances, and Schur square dimensions from these matrices using standard properties of algebraic geometry codes. These steps invoke external results on divisors, differentials, and code parameters over finite fields rather than fitting parameters to data, redefining inputs as outputs, or chaining self-citations that collapse the claims. No load-bearing step reduces by construction to the paper's own assumptions or prior outputs; the central claims remain independently verifiable from the given curve and twist data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard domain assumptions from algebraic geometry and coding theory; no free parameters or invented entities introduced in the abstract.

axioms (1)
  • domain assumption Weil differentials on elliptic curves over finite fields yield explicit parity-check matrices for the twisted codes.
    Invoked to construct the parity-check matrices as stated in the abstract.

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Twisted Reed–Solomon Codes

    P. Beelen, S. Punchinger and J. Rosenkilde, “Twisted Reed–Solomon Codes”, IEEE Trans. Inf. Theory, vol. 68, no. 5, pp. 3047-3061, May 2022

    work page 2022

  2. [2]

    Many Non-Reed-Solomon Type MDS Codes From Arbitray Genus Algebraic Curves

    H. Chen, “Many Non-Reed-Solomon Type MDS Codes From Arbitray Genus Algebraic Curves”, IEEE Trans. Inf. Theory, vol. 70, no. 7, pp. 4856-4864, Jul. 2024

    work page 2024

  3. [3]

    Hard Problems of Algebraic Geometry Codes

    Q. Cheng, “Hard Problems of Algebraic Geometry Codes”, IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 402-406, Jan. 2008

    work page 2008

  4. [4]

    Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and Their Subcodes

    A. Couvreur, I. M ´arquez-Corbella and R. Pellikaan, “Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and Their Subcodes”, IEEE Trans. Inf. Theory, vol. 63, no. 8, pp. 5404-5418, Aug. 2017

    work page 2017

  5. [5]

    Deep Holes of Twisted Reed-Solomon Codes

    W. Fang and J. Xu, “Deep Holes of Twisted Reed-Solomon Codes”, 2024 IEEE International Symposium on Information Theory (ISIT), Athens, Greece, pp. 488-493, 2024

    work page 2024

  6. [6]

    On Twisted Generalized Reed-Solomon Codes WithℓTwists

    H. Gu and J. Zhang, “On Twisted Generalized Reed-Solomon Codes WithℓTwists”, IEEE Trans. Inf. Theory, vol. 70, no 1, pp. 145-153, Jan. 2024

    work page 2024

  7. [7]

    Design and Implementation of an Efficient Elliptic Curve Digital Signature Algorithm (ECDSA),

    Y . Genc ¸ and E. Afacan, “Design and Implementation of an Efficient Elliptic Curve Digital Signature Algorithm (ECDSA),” 2021 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), Toronto, ON, Canada, pp. 1-6, 2021

    work page 2021

  8. [8]

    Stichtenoth, Algebraic Function Fields and Codes, vol

    H. Stichtenoth, Algebraic Function Fields and Codes, vol. 254, 2nd ed. Berlin, Germany: Springer-Verlag, 2009

    work page 2009

  9. [9]

    Self-dual Goppa Codes

    H. Stichtenoth, “Self-dual Goppa Codes”, J. Pure, Appl. Algebra vol. 55, nos. 1-2, pp. 199-211, Nov. 1998

    work page 1998

  10. [10]

    On(L,P)−Twisted Generalized Reed-Solomon Codes

    Z. Hu, L. Wang, N. Li, X. Zeng and X. Tang, “On(L,P)−Twisted Generalized Reed-Solomon Codes”[Online]. Available: https://arxiv.org /abs/2502.04746v1, Feb 2025. November 26, 2025 DRAFT 41

    work page arXiv 2025

  11. [11]

    A Tight Upper Bound for the Maximal Length of MDS Elliptic Codes

    D. Han and Y . Ren, “A Tight Upper Bound for the Maximal Length of MDS Elliptic Codes”, IEEE Trans. Inf. Theory, vol. 69, no. 2, pp. 819-822, Feb. 2023

    work page 2023

  12. [12]

    The Maximal Length of q-ary MDS Elliptic Codes Is Close toq/2

    D. Han and Y . Ren, “The Maximal Length of q-ary MDS Elliptic Codes Is Close toq/2”, International Mathematics Research Notices, vol. 2024, no. 11, pp: 9036–9043, June 2024

    work page 2024

  13. [13]

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

    D. Huang, Q. Yue and Y . Niu “MDS or NMDS LCD codes from twisted genralized Reed-Solomon codes”. Des. Codes Cryptogr. vol. 15, no. 2, pp. 221-237, March 2023

    work page 2023

  14. [14]

    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. Inf. Theory, doi: 10.1109/TIT.2024.3509895

    work page doi:10.1109/tit.2024.3509895 2024

  15. [15]

    Self-Dual Near MDS Codes from Elliptic Curves

    L. Jin and H. Kan, “Self-Dual Near MDS Codes from Elliptic Curves”, IEEE Trans. Inf. Theory, vol. 65, no. 4, pp. 2166-2170, Apr. 2019

    work page 2019

  16. [16]

    New families of non-Reed-Solomon MDS codes

    L. Jin, L. Ma, C. Xing and H. Zhou, “New families of non-Reed-Solomon MDS codes”[Online]. Avail- able:https://arxiv.org/abs/2411.14779v1, Nov. 2024

    work page arXiv 2024

  17. [17]

    Goppa-Like AG Codes FromC a,b Curves and Their Behavior Under Squaring Their Dual

    S. Khalfaoui, M. Lhotel and J. Nardi, “Goppa-Like AG Codes FromC a,b Curves and Their Behavior Under Squaring Their Dual”, IEEE Trans. Inf. Theory, vol. 70, no. 5, pp. 3330-3344, May. 2024

    work page 2024

  18. [18]

    Optimal Locally Repairable Codes via Elliptic Curves

    X. Li, L. Ma, C. Xing, “Optimal Locally Repairable Codes via Elliptic Curves”, IEEE Trans. Inf. Theory, vol. 65, no. 1, pp. 108-117, Jan. 2019

    work page 2019

  19. [19]

    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”. Des. Codes Cryptogr. vol. 88, no. 7, pp. 1285-1300, July 2020

    work page 2020

  20. [20]

    On the minimum distance of elliptic curve codes

    J. Li,D. Wan and J. Zhang, “On the minimum distance of elliptic curve codes”, Proc. IEEE Int. Symp. Inf. Theory (ISIT), pp. 2391–2395, Jun. 2015

    work page 2015

  21. [21]

    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. Inf. Theory, doi: 10.1109/TIT.2025.3541799, Feb. 2025

    work page doi:10.1109/tit.2025.3541799 2025

  22. [22]

    Performance Comparison Between RSA and Elliptic Curve Cryptography-Based QR Code Authentication

    N. Thiranant, Y . Lee, H. Lee, “Performance Comparison Between RSA and Elliptic Curve Cryptography-Based QR Code Authentication”, 2015 IEEE 29th International Conference on Advanced Information Networking and Applications Workshops, Gwangju, Korea (South), pp. 278-282, 2015

    work page 2015

  23. [23]

    Two classes of twisted generalized Reed-Solomon codes with two twists

    S. Yang, J. Wang and Y . Wu, “Two classes of twisted generalized Reed-Solomon codes with two twists”, Finite Fields. and Appl., vol. 104, https://doi.org/10.1016/j.ffa.2025.102595, Feb 2025

    work page doi:10.1016/j.ffa.2025.102595 2025

  24. [24]

    On Deep Holes of Elliptic Curve Codes

    J. Zhang and D. Wan, “On Deep Holes of Elliptic Curve Codes”, IEEE Trans. on Inf Theory, vol. 69, no. 7, pp. 4498-4506, Jul. 2023

    work page 2023

  25. [25]

    A class of twisted generalized Reed-Solomon codes

    J. Zhang, Z. Zhou and C. Tang, “A class of twisted generalized Reed-Solomon codes”, Des, Codes and Cryptogr, vol. 90, no. 7, pp. 1649-1658, July 2022. November 26, 2025 DRAFT

    work page 2022