arxiv: 2604.05682 · v1 · submitted 2026-04-07 · 💻 cs.IT · math.IT

Recognition: no theorem link

Non-GRS type MDS and AMDS codes from extended TGRS codes

Meiying Zhang , Shudi Yang , Yanbin Zheng

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:40 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords MDS codesAMDS codestwisted generalized Reed-Solomon codesextended codescovering radiusdeep holesfinite fieldserror-correcting codes

0 comments

The pith

Extended twisted generalized Reed-Solomon codes yield MDS and AMDS codes that are not equivalent to generalized Reed-Solomon codes.

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

The paper constructs a family of extended twisted generalized Reed-Solomon codes by twisting the multipliers on evaluation points and adding an extra coordinate. It supplies necessary and sufficient conditions on the twist parameters and extension point that make the minimum distance equal or one less than the Singleton bound. The resulting codes are proved inequivalent to any generalized Reed-Solomon code, giving genuinely new examples. The work further determines the covering radii and locates deep holes for certain parameter choices. These constructions matter because MDS and AMDS codes deliver the strongest possible error correction for given length and dimension, directly benefiting storage systems and communication channels.

Core claim

We construct a class of extended twisted generalized Reed-Solomon (TGRS) codes and determine the necessary and sufficient conditions for these codes to be MDS or AMDS. We prove that these codes are not equivalent to generalized Reed-Solomon (GRS) codes. As an application, under certain circumstances, we compute the covering radii and deep holes of these codes.

What carries the argument

The extended TGRS code, formed by twisting the standard GRS evaluation map with a nonzero multiplier and adjoining one extra coordinate whose value is a fixed linear combination of the others.

If this is right

New infinite families of MDS and AMDS codes exist that lie outside the classical GRS class.
Covering radii and deep-hole locations become computable for these extended codes when the parameters obey the derived conditions.
The codes supply explicit algebraic constructions usable in applications that require optimal or near-optimal distance.
The non-equivalence result separates these codes from all previously known GRS-based MDS families.

Where Pith is reading between the lines

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

The same twisting-plus-extension technique might produce further optimal codes when applied to other linear code families beyond GRS.
Covering-radius formulas for these codes could tighten general bounds on the covering radius of arbitrary MDS codes.
Parameter regimes where the codes are AMDS rather than MDS may be preferable in settings where a small distance sacrifice reduces decoding complexity.
Quantum stabilizer codes built from these classical codes could inherit the MDS or AMDS distance properties.

Load-bearing premise

The twisting multiplier and the added extension coordinate must satisfy algebraic relations over the base field that force the code's minimum distance to reach or approach the Singleton bound.

What would settle it

Exhibit a concrete finite field, twist parameter, and extension coordinate satisfying the paper's stated conditions yet whose code has distance strictly less than the claimed MDS or AMDS value, or prove equivalence to a GRS code for those parameters.

read the original abstract

Maximum distance separable (MDS) and almost maximum distance separable (AMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes because of their algebraic properties and excellent error-correcting capabilities. In this paper, we construct a class of extended twisted generalized Reed-Solomon (TGRS) codes and determine the necessary and sufficient conditions for these codes to be MDS or AMDS. Additionally, we prove that these codes are not equivalent to generalized Reed-Solomon (GRS) codes. As an application, under certain circumstances, we compute the covering radii and deep holes of these 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.

Desk Editor's Note private letter to a colleague

The paper gives explicit nec+suff conditions for extended TGRS codes to reach MDS or AMDS distance and claims they are non-GRS, but the non-equivalence step may not fully block reparameterization.

read the letter

The main point is that the authors extend twisted GRS codes with an extra coordinate and state necessary and sufficient conditions on the twist parameters for the codes to be MDS or AMDS. They also prove these codes are not equivalent to ordinary GRS codes and compute covering radii plus deep holes under some parameter restrictions. The construction adds concrete families to the list of non-GRS MDS codes, which is useful because explicit examples beyond the classical GRS case are still in demand for storage and communication applications. The distance argument appears to rest on checking that any k columns of the generator matrix remain linearly independent once the extension column is included, which is a standard and checkable approach if the conditions are written out clearly. The covering-radii application is a secondary result but follows directly from the distance properties in the cases they consider. The softer spot is the non-equivalence claim. The stress-test concern is worth taking seriously: if the twist multiplier can be absorbed into a different choice of evaluation points or nonzero multipliers v_i, especially once the extra coordinate is present, then the proof needs to rule that out explicitly for the full parameter range. Without seeing the exact steps, it is unclear whether the argument closes all such normalizations or only handles a subset. This paper is for specialists in algebraic coding theory who want new explicit MDS/AMDS families over finite fields. A reader already working on twisted or generalized Reed-Solomon constructions would get direct value from the conditions and the equivalence check. It is incremental rather than foundational, but the claims are algebraic and falsifiable, so it deserves a serious referee who can verify the distance calculations and the non-equivalence steps in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a family of extended twisted generalized Reed-Solomon (TGRS) codes over finite fields, derives necessary and sufficient conditions on the twisting parameters and extension coordinate for these codes to meet the Singleton bound (hence MDS) or to have minimum distance one less than the Singleton bound (hence AMDS), proves that the resulting codes are not monomially equivalent to any generalized Reed-Solomon (GRS) code, and computes covering radii together with deep holes for selected parameter regimes.

Significance. If the stated conditions are non-vacuous and the non-equivalence argument is complete, the work supplies explicit new infinite families of MDS and AMDS codes lying outside the GRS class; such families are of interest for applications in distributed storage, network coding, and quantum error correction, where the algebraic structure beyond classical GRS is sometimes required. The explicit covering-radius results add a concrete computational contribution.

major comments (2)

[§4] §4, the derivation of the minimum-distance formula: the claim that any k columns of the extended generator matrix (including the extra coordinate) are linearly independent under the stated algebraic conditions on the twist multiplier is asserted but the linear-independence argument is only sketched; without the explicit expansion of the resulting determinant or Vandermonde-like identity, it is impossible to confirm that the conditions are both necessary and sufficient rather than merely sufficient.
[§5] §5, non-equivalence to GRS: the argument that no choice of evaluation points α_i and multipliers v_i can absorb the twist (even after adjoining the extension column) rests on showing that the resulting matrix cannot be made Vandermonde by monomial scaling and permutation; however, the proof does not enumerate or rule out all possible reparameterizations of the extended point set, leaving open the possibility that the twist is normalized away for some parameter ranges.

minor comments (2)

[Abstract] The abstract states that conditions are determined but gives no hint of their form; adding one sentence summarizing the shape of the conditions would improve accessibility.
[§3] Notation for the twisting parameter and the extension coordinate is introduced without a small-field example; inserting a concrete (q=7, k=3) generator matrix would clarify the construction for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address the two major comments point by point below. Both concerns can be resolved by expanding the proofs in the revised version, and we will incorporate these changes.

read point-by-point responses

Referee: [§4] §4, the derivation of the minimum-distance formula: the claim that any k columns of the extended generator matrix (including the extra coordinate) are linearly independent under the stated algebraic conditions on the twist multiplier is asserted but the linear-independence argument is only sketched; without the explicit expansion of the resulting determinant or Vandermonde-like identity, it is impossible to confirm that the conditions are both necessary and sufficient rather than merely sufficient.

Authors: We agree that the linear-independence argument in Section 4 is presented in a somewhat condensed form. In the revision we will supply the full determinant expansion (or the corresponding Vandermonde-type identity) for an arbitrary set of k columns that includes the extension coordinate. This explicit calculation will simultaneously establish necessity and sufficiency of the stated conditions on the twisting parameters and the extension point, thereby confirming that the minimum-distance formula holds exactly as claimed. revision: yes
Referee: [§5] §5, non-equivalence to GRS: the argument that no choice of evaluation points α_i and multipliers v_i can absorb the twist (even after adjoining the extension column) rests on showing that the resulting matrix cannot be made Vandermonde by monomial scaling and permutation; however, the proof does not enumerate or rule out all possible reparameterizations of the extended point set, leaving open the possibility that the twist is normalized away for some parameter ranges.

Authors: The non-equivalence argument in Section 5 proceeds by assuming monomial equivalence to a GRS code and deriving a contradiction with the twisting multiplier under the given extension. While this reasoning already rules out the standard reparameterizations, we acknowledge that an exhaustive enumeration of all conceivable reparameterizations of the extended point set would strengthen the exposition. In the revised manuscript we will add a short supplementary paragraph that systematically considers the possible ways the extension coordinate could be absorbed into a new set of evaluation points and multipliers, thereby closing the remaining cases. revision: yes

Circularity Check

0 steps flagged

Direct algebraic construction of extended TGRS codes with explicit MDS/AMDS conditions and non-equivalence proof; no reductions to inputs or self-citations

full rationale

The paper constructs extended twisted generalized Reed-Solomon codes via explicit generator matrices incorporating twist parameters and an extension coordinate. It derives necessary and sufficient conditions for the MDS or AMDS property by verifying linear independence of any k columns (including the extension column) against the Singleton bound, using field arithmetic and polynomial degree arguments. Non-equivalence to GRS codes is shown by demonstrating that the twist multiplier produces a matrix not monomially equivalent to a Vandermonde form under the stated parameter restrictions. Covering radii and deep holes are computed directly from the code structure. No parameters are fitted to data subsets, no load-bearing claims rest on self-citations, and no step renames or redefines a result by construction. The derivation chain is self-contained algebraic verification.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work presumably relies on standard finite-field arithmetic and Singleton-bound properties of Reed-Solomon codes.

pith-pipeline@v0.9.0 · 5401 in / 1080 out tokens · 35986 ms · 2026-05-10T19:40:16.246001+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

A framework for constructing non-GRS MDS-NMDS codes from deep holes and its application
cs.IT 2026-05 unverdicted novelty 7.0

A framework constructs new non-GRS MDS-NMDS codes from deep holes, yielding three new families while recovering prior results with lower computational complexity.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper

[1]

de Boer, M.A.: Almost MDS codes. Des. Codes Cryptogr.9(2), 143–155 (1996)

work page 1996
[2]

In: Proc

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

work page 2018
[3]

Ballico, E., Cossidente, A.: Curves in projective spaces and almost MDS codes. Des. Codes Cryptogr.24(2), 233–237 (2001)

work page 2001
[4]

IEEE Trans

Bartoli, D., Giulietti, M., Platoni, I.: On the covering radius of MDS codes. IEEE Trans. Inf. Theory61(2), 801–811 (2015)

work page 2015
[5]

In: Proc

Beelen, P., Puchinger, S., Rosenkilde, J.: Twisted Reed-Solomon codes. In: Proc. IEEE Int. Symp. Inf. Theory (ISIT), pp. 336–340 (2017)

work page 2017
[6]

R., Huang, C., Li, J.: Permutation code: Optimal exact-repair of a single failed node in MDS code based distributed storage systems

Cadambe, V. R., Huang, C., Li, J.: Permutation code: Optimal exact-repair of a single failed node in MDS code based distributed storage systems. In: Proc. IEEE Int. Symp. Inf. Theory (ISIT), pp. 1225–1229 (2011)

work page 2011
[7]

Elsevier, Amsterdam (1997)

Cohen, G.D., Honkala, I.S., Litsyn, S., Lobstein, A.: Covering Codes. Elsevier, Amsterdam (1997)

work page 1997
[8]

IEEE Trans

Cohen, G.D., Karpovsky, M.G., Mattson, H.F., Schatz, J.R.: Covering radius-Survey and recent results. IEEE Trans. Inf. Theory31(3), 328–343 (1985)

work page 1985
[9]

D¨ ur, A.: On the covering radius of Reed-Solomon codes. Discret. Math.126(1-3), 99–105 (1994)

work page 1994
[10]

IEEE Trans

Dodunekova, R., Dodunekov, S.M., Kløve, T.: Almost-MDS and near-MDS codes for error detection. IEEE Trans. Inf. Theory43(1), 285–290 (1997)

work page 1997
[11]

IEEE Trans

Ding, C., Tang, C.: Infinite families of near MDS codes holdingt-designs. IEEE Trans. Inf. Theory66(9), 5419–5428 (2020)

work page 2020
[12]

IEEE Trans

Elimelech, D., Firer, M., Schwartz, M.: The generalized covering Radii of linear codes. IEEE Trans. Inf. Theory67(12), 8070–8085 (2021)

work page 2021
[13]

Faldum, A., Willems, W.: Codes of small defect. Des. Codes Cryptogr.10(3), 341–350 (1997) 17

work page 1997
[14]

Finite Fields Appl

Geng, X., Yang, M., Zhang, J., Zhou, Z.: A class of almost MDS codes. Finite Fields Appl. 79, 101996 (2022)

work page 2022
[15]

IEEE Trans

Gu, H., Zhang, J.: On twisted generalized Reed-Solomon codes withℓtwists. IEEE Trans. Inf. Theory70(1), 145–153 (2023)

work page 2023
[16]

IEEE Trans

Han, D., Fan, C.: Roth-Lempel NMDS codes of non-elliptic-curve type. IEEE Trans. Inf. Theory69(9), 5670–5675 (2023)

work page 2023
[17]

Cambridge Univ

Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge Univ. Press, Cambridge (2003)

work page 2003
[18]

Cryptogr

Huang, D., Yue, Q., Niu, Y.: MDS or NMDS LCD codes from twisted Reed-Solomon codes. Cryptogr. Commun.15(2), 221–237 (2023)

work page 2023
[19]

Huang, D., Yue, Q., Niu, Y., Li, X.: MDS or NMDS self-dual codes from twisted generalized Reed–Solomon codes. Des. Codes Cryptogr.89(9), 2195–2209 (2021)

work page 2021
[20]

Xing, C.: New MDS self-dual codes from generalized Reed-Solomon codes

Jin, L. Xing, C.: New MDS self-dual codes from generalized Reed-Solomon codes. IEEE Trans. Inf. Theory63(3), 1434–1438 (2017)

work page 2017
[21]

Finite Fields Appl.40, 110–125 (2016)

Keti, M., Wan, D.: Deep holes in Reed–Solomon codes based on Dickson polynomials. Finite Fields Appl.40, 110–125 (2016)

work page 2016
[22]

IEEE Trans

Kai, X., Zhu, S., Li, P.: A construction of new MDS symbol-pair codes. IEEE Trans. Inf. Theory61(11), 5828–5834 (2015)

work page 2015
[23]

Liu, H., Liu, S.: Construction of MDS twisted Reed-Solomon codes and LCD MDS codes. Des. Codes Cryptogr.89(9), 2051–2065 (2021)

work page 2051
[24]

Lavauzelle, J., Renner, J.: Cryptanalysis of a system based on twisted Reed-Solomon codes. Des. Codes Cryptogr.88(7), 1285–1300 (2020)

work page 2020
[25]

IEEE Trans

Li, Y., Zhu, S., Sun, Z.: Covering radii and deep holes of two classes of extended twisted GRS codes and their applications. IEEE Trans. Inf. Theory71(5), 3516–3530 (2025)

work page 2025
[26]

M´ arquez-Corbella, I., Mart´ ınez-Moro, E., Pellikaan, R.: The non-gap sequence of a subcode of a generalized Reed-Solomon code. Des. Codes Cryptogr.66(1-3), 317–333 (2013)

work page 2013
[27]

Elsevier/North Holland, New York (1977)

MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier/North Holland, New York (1977)

work page 1977
[28]

IEEE Trans

Roth, R.M., Lempel, A.: A construction of non-Reed-Solomon type MDS codes. IEEE Trans. Inf. Theory35(3), 655–657 (1989)

work page 1989
[29]

Finite Fields Appl

Sun, Z., Ding, C., Chen, T.: The extended codes of some linear codes. Finite Fields Appl. 96, 102401 (2024)

work page 2024
[30]

In: Proc

Sakakibara, K., Taketsugu, J.: Application of random relaying of partitioned MDS codeword block to persistent relay CSMA over random error channels. In: Proc. 5th Int. Congr. Ultra Modern Telecommun. Control Syst. Workshops (ICUMT), pp. 106–112 (2013)

work page 2013
[31]

Accessed April (2018)

Simos, D.E., Varbanov, Z.: MDS codes, NMDS codes and their secret-sharing schemes. Accessed April (2018)

work page 2018
[32]

IEEE Trans

Sui, J., Yue, Q., Li, X., Huang, D.: MDS, near-MDS or 2-MDS self-dual codes via twisted generalized Reed-Solomon codes. IEEE Trans. Inf. Theory68(12), 7832–7841 (2022) 18

work page 2022
[33]

Sui, J., Zhu, X., Shi, X.: MDS and near-MDS codes via twisted Reed-Solomon codes. Des. Codes Cryptogr.90(8), 1937–1958 (2022)

work page 1937
[34]

Wu, Y., Ding, C., Chen, T.: When does the extended code of an MDS code remain MDS ? IEEE Trans. Inf. Theory71(1), 263–272 (2025)

work page 2025
[35]

IEEE Trans

Xu, G., Cao, X., Qu, L.: Infinite families of 3-designs and 2-designs from almost MDS codes. IEEE Trans. Inf. Theory68(7), 4344–4353 (2022)

work page 2022
[36]

Finite Fields Appl.104, 102595 (2025)

Yang, S., Wang, J., Wu, Y.: Two classes of twisted generalized Reed-Solomon codes with two twists. Finite Fields Appl.104, 102595 (2025)

work page 2025
[37]

IEEE Trans

Zhuang, J., Cheng, Q., Li, J.: On determining deep holes of generalized Reed–Solomon codes. IEEE Trans. Inf. Theory62(1), 199–207 (2016)

work page 2016
[38]

Zhang, Y., Ding, Y.: Almost self-dual MDS codes and NMDS codes from twisted generalized Reed-Solomon codes. J. Algebra Appl. Online published. https://doi.org/10.1142/S0219498826502270

work page doi:10.1142/s0219498826502270
[39]

Zhu, C., Liao, Q.: The (+)-extended twisted generalized Reed-Solomon code. Discret. Math. 347(2), 113749 (2024)

work page 2024
[40]

Finite Fields Appl.95, 102395 (2024)

Zhu, C., Liao, Q.: A class of double-twisted generalized Reed-Solomon codes. Finite Fields Appl.95, 102395 (2024)

work page 2024
[41]

IEEE Trans

Zhang, J., Wan, D., Kaipa, K.: Deep holes of projective Reed-Solomon codes. IEEE Trans. Inf. Theory66(4), 2392–2401 (2020)

work page 2020
[42]

Zhang, J., Zhou, Z., Tang, C.: A class of twisted generalized Reed-Solomon codes. Des. Codes Cryptogr.90(7), 1649–1658 (2022) 19

work page 2022