The error-correcting pair for several classes of NMDS linear codes

Dong He; Qunying Liao; Zhaohui Zhang

REVIEW 3 major objections 5 minor 25 references

The error-correcting pair for several classes of NMDS linear codes

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper shows that, under the stated length constraints, NMDS linear codes with an $\ell$-error-correcting pair can realize several candidate parameter shapes only in the degenerate case $C=[n,2,n-2]$ (with $n$ odd or even according to…

desk verdict Solid incremental work on error-correcting pairs for NMDS codes; send to a referee after asking for an explicit scalar-extension caveat. read the letter →

arxiv 2506.07380 v1 pith:Y646RWV6 submitted 2025-06-09 cs.IT math.IT

Dong He , Zhaohui Zhang , Qunying Liao This is my paper

classification cs.ITmath.IT MSC 94B0594B3594B27

keywords error-correctingpairNMDScodesProductSingletonBoundSchurPMDStwistedgeneralizedReed-SolomonMDSdecoding

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

An $\ell$-error-correcting pair is a pair of auxiliary linear codes $(A,B)$ whose Schur product lies in $C^\perp$; when it exists, $C$ can be decoded from up to $\ell$ errors in polynomial time. This paper asks which NMDS codes, near-MDS codes useful in secret sharing and communication, can carry such a pair. Using the Product Singleton Bound, the authors show that for an NMDS code of odd minimum distance $2\ell+1$, the candidate parameter shapes $A=[n,\ell+3,n-\ell-2]$ and $A=[n,\ell+2,n-\ell-1]$ are possible only in the extreme case $C=[n,2,n-2]$ with $n$ odd. They give the analogous collapse for even minimum distance $2\ell+2$, and for the remaining candidate shapes they list exactly what $B^\perp$ and the Schur product $A\ast B$ may look like. Concrete examples built from twisted generalized Reed-Solomon codes realize the first cases, and a strengthened proof shows the MDS version of the classification does not need the large-field condition used earlier.

What carries the argument

The load-bearing object is the Schur product code $A\ast B$ together with the Product Singleton Bound $d(A\ast B)\le\max\{1,n-k(A)-k(B)+2\}$. Because an error-correcting pair requires $A\ast B\subseteq C^\perp$, any lower bound on $d(A\ast B)$ inherited from $d(C^\perp)$ must fit inside this upper bound, and when the two meet, $(A,B)$ is a Product-MDS pair whose components are MDS or generalized Reed-Solomon codes with a common evaluation-point sequence. The proofs combine this with a distance lower bound from dual distances, full-support properties of MDS codes, and the dimension formula $k(A\ast B)\ge k(A)+k(B)-1$ for full-support MDS factors. Twisted generalized Reed-Solomon codes, defined by evaluating twisted polynomials, supply the NMDS examples that realize the first parameter cases.

What would settle it

Search for a $q$-ary NMDS code $C=[n,n-2\ell-1,2\ell+1]$ with $2\le\ell<n/2-1$ and $\ell\ne(n-3)/2$ that has an $\ell$-error-correcting pair $(A,B)$ over $F_{q^m}$ with $A=[n,\ell+3,n-\ell-2]$ or $A=[n,\ell+2,n-\ell-1]$; any such code directly contradicts Theorems 3.1 and 3.2. A small computer search over $q$, $n$, and $\ell$ would settle the question, and the same search should check whether computing $A\ast B$ over $F_{q^m}$ rather than $F_q$ changes the Product Singleton upper bound used in the proofs.

Watch

Extended reading notes

Core claim

On the paper's own terms: if $C=[n,n-2\ell-1,2\ell+1]$ is NMDS and has an $\ell$-error-correcting pair $(A,B)$ over $F_{q^m}$, then when $A$ has parameters $[n,\ell+3,n-\ell-2]$ or $[n,\ell+2,n-\ell-1]$, the length constraints force $\ell=(n-3)/2$, so $C=[n,2,n-2]$ and $n$ is odd. When $A=[n,\ell+1,n-\ell]$, the parameters of $B^\perp$ and $A\ast B$ are restricted to three explicit possibilities: $B^\perp=[n,n-\ell-1,\ell+1]$, or $B^\perp=[n,n-\ell,\ell+1]$ together with $A\ast B=[n,2\ell,n-2\ell+1]$, or $B^\perp=[n,n-\ell,\ell+1]$ together with $A\ast B=C^\perp$. For even minimum distance $2\ell+2$, the same pattern holds: cases $[n,\ell+4,n-\ell-3]$ and $[n,\ell+3,n-\ell-2]$ force $C=[n,2,n-2]$ with $n$ even, and $[n,\ell+2,n-\ell-1]$ gives three listed possibilities involving $B^\perp$ and $A\ast B$. The paper also proves a field-size-free version of earlier MDS results: when $C$ is MDS of even minimum distance and $A$ is one of the two largest candidate shapes, $A$, $B$, and $C$ are generalized Reed-Solomon codes with a common evaluation-point sequence and $B=(A\ast C)^\perp$.

Load-bearing premise

The arguments assume that the Product Singleton Bound and the PMDS lemmas, which are stated for $F_q$-linear codes, carry over unchanged to the setting where $A$ and $B$ are linear over the extension field $F_{q^m}$ while only $C$ is defined over $F_q$; dimension equalities such as $k(A\ast B)=k(C^\perp)$ require the dimensions to be taken over the same field.

Editorial extensions

If this is right

For odd-distance NMDS codes, among the cases this paper treats, the shape $A=[n,\ell+1,n-\ell]$ escapes the dimension-two collapse, while the $A=[n,\ell+3,n-\ell-2]$ and $A=[n,\ell+2,n-\ell-1]$ shapes are confined to $C=[n,2,n-2]$ with $n$ odd.
For even-distance NMDS codes, the analogous survivor is $A=[n,\ell+2,n-\ell-1]$, while $A=[n,\ell+4,n-\ell-3]$ and $A=[n,\ell+3,n-\ell-2]$ force $C=[n,2,n-2]$ with $n$ even.
In the surviving cases, the Schur product $A\ast B$ is either exactly $C^\perp$ or one explicitly listed near-MDS code, so any NMDS code admitting such a pair has a highly structured dual and auxiliary pair.
The earlier MDS classification for even minimum distance remains valid without the condition $q^m>\max_i\binom{n}{i}$, so the generalized Reed-Solomon conclusion holds for all extension-field sizes.
Twisted generalized Reed-Solomon constructions give explicit small-field examples over $F_{37}$ with parameters $[10,3,7]$ and $[10,4,6]$, showing the listed restrictions are not vacuous.

Reading between the lines

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

A natural next step is to try to rule out the untreated NMDS parameter shapes, such as $A=[n,\ell+1,n-\ell-1]$, $A=[n,\ell+2,n-\ell-2]$, and their even-distance analogues, with the same Product Singleton machinery; if they collapse too, the class of NMDS codes with error-correcting pairs would be essentially exhausted by the surviving shapes.
The proofs transfer lemmas stated over $F_q$ to pairs over $F_{q^m}$; checking small examples directly over extension fields would confirm that field-degree factors do not change the listed parameter triples.
Since several listed possibilities force $A\ast B=C^\perp$, NMDS codes admitting error-correcting pairs are Schur-product-dual to their auxiliary pair, which may make their decoding complexity and their secret-sharing access structures more tractable than generic NMDS codes.
The two $q=37$ examples suggest a family: twisted generalized Reed-Solomon codes with the same hook value and evaluation set realize the first cases for different dimensions; varying $n$ and $\ell$ could produce more examples or expose limitations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

It's a careful, narrow case analysis that does what it says: it narrows the parameter search for error-correcting pairs of NMDS codes, and it deserves a serious referee. The new necessary conditions for cases A.1, A.2, A.4, A.5 and D.2, D.4, D.7, D.8 are real, and the Section 5 proof—which removes the q^m > max binomial condition from Lemmas 2.2–2.3—is a genuine simplification. The two TGRS examples are small and checkable by hand.

The main strategy is straightforward: combine the Product Singleton Bound and PMDS lemmas with the case enumeration from the authors' earlier paper [7]. I read Theorems 3.1–3.3 and 4.1–4.3; the dimension arguments hold up. The perceived "gap" between the stated range and the stricter range used in the contradiction is mostly a wording issue. The proof assumes ℓ < (n−3)/2, gets a contradiction, and the boundary case ℓ = (n−3)/2 gives C = [n,2,n−2] by parameter substitution. Logical, though the exposition could be clearer.

The real soft spot is the extension-field convention. Definition 2.1 says A and B are F_{q^m}-linear and then requires A*B ⊆ C^⊥, where C^⊥ is the q-ary dual. As written, that is only meaningful if you extend scalars to F_{q^m}. Without that extension, E.1 is vacuous for m>1; with it, all the dimension comparisons in the theorems go through unchanged. This is a clarity issue, not a mathematical error—the standard definition in [14] implicitly uses the extended dual—but the authors should state it explicitly in Section 2. It's the one thing I'd insist on before publication. The corollaries that say "same proof" also omit details, which is a minor irritant but not a correctness problem.

The self-citation pattern is fine: [7] is their own earlier enumeration, but they are deriving new restrictions from it, and Section 5 gives an independent proof that does not rely on those lemmas. No circularity.

Who is this for? Researchers working on error-correcting pairs or NMDS codes. It's not a field-shifting paper, but it's a solid, verifiable increment. I'd send it to a competent referee. I wouldn't bring it to a general reading group, though a specialized one might want to check the examples and the field-transfer point.

Referee Report

3 major / 5 minor

Summary. The paper studies error-correcting pairs (ECPs) for near maximum distance separable (NMDS) linear codes. Building on the authors' earlier enumeration of possible parameters of A for NMDS codes with minimum distance 2ℓ+1 (six cases) and 2ℓ+2 (ten cases), it uses the Product Singleton Bound and PMDS theory to give necessary conditions on B^⊥ and A*B for several of these cases, and supplies TGRS-based examples for case A.1 (odd distance) and cases D.1/D.2 (even distance). It also presents a proof of two earlier MDS results of [5,24], claiming to remove a large-field condition. The main theorems assert, for specific parameter cases of A, that either certain parameter triples for B^⊥ and A*B must occur, or that the code must be of the form [n,2,n−2] with n of the appropriate parity.

Significance. If the central claims are correct, the paper gives the first structural restrictions on the possible error-correcting pairs for several families of NMDS codes, a natural counterpart to the MDS classification of Márquez-Corbella and Pellikaan. The TGRS examples are concrete and the Section 5 derivation of the MDS GRS conclusions without the q^m > max binomial condition is a useful simplification. The main obstacle is a foundational field-extension convention: the proofs mix dimensions over F_q and F_{q^m}, and the definition of E.1 is internally inconsistent unless C^⊥ is explicitly extended. Once that convention is fixed, the dimension arguments in the core theorems are largely sound, but the current manuscript is not self-consistent as written.

major comments (3)

[Sec. 2, Def. 2.1 (E.1) and Sec. 2.4] As written, E.1 is inconsistent for m>1: A and B are F_{q^m}-linear, so A*B is an F_{q^m}-subspace of F_{q^m}^n, whereas C^⊥ is defined in the Preliminaries as a subspace of F_q^n. A nonzero extension-linear subspace cannot be contained in F_q^n, so E.1 either forces A*B = 0 or must be read with an implicit scalar extension of C^⊥. Every subsequent dimension comparison—for example k(A*B) > k(C^⊥) in Theorems 3.1–3.3 and 4.1–4.3, and the equalities A*B = C^⊥ in Theorems 3.3 and 4.3—mixes the F_{q^m}-dimension of A*B with the F_q-dimension of C^⊥. The manuscript must explicitly define C^⊥ after extension to F_{q^m} (e.g., C^⊥ ⊗_{F_q} F_{q^m}), prove that d(C^⊥) and the dimension n−k(C) are preserved, and restate Lemmas 2.6–2.9 and Propositions 2.1–2.2 for that extended setting. Without this convention the main theorems are vacuous as stated.
[Theorems 3.1, 3.2, 4.1, 4.2] Each of these proofs begins with a stricter range than the theorem statement. For example, Theorem 3.1 states 2 ≤ ℓ < n/2 − 1 but the proof takes 2 ≤ ℓ < (n−3)/2, and Theorem 4.2 similarly restricts to 2 ≤ ℓ < n/2 − 2 before deriving a contradiction. After the contradiction the proof merely asserts 'From the above' that the remaining boundary value is the only possibility, without spelling out the case analysis. For even n the complementary interval is empty, so the claimed conclusions 'n is odd' or 'n is even' are only vacuously true and should be stated as such. The boundary cases need a separate explicit argument, or the theorem statements should be restricted to the range actually proved.
[Sec. 2.4, application of PMDS lemmas] Lemma 2.7, Lemma 2.8, Lemma 2.9, Proposition 2.1 and Proposition 2.2 are stated for linear codes over a single field F_q, but in the main proofs they are applied to A and B over F_{q^m} while C and C^⊥ are over F_q. Since the field-extension convention of C^⊥ is not defined, the PMDS conclusions 'A*B is MDS' and 'A, B, A*B are GRS' are not automatically valid in the mixed-field setting. This affects the central arguments of Theorems 3.1, 3.2, 3.3, 4.1, 4.2 and 4.3, because those theorems rely on the dimension and distance equalities obtained from the PMDS machinery.

minor comments (5)

[Corollaries 3.1 and 4.1] These corollaries are stated as consequences 'in the same proof as' Theorems 3.2 and 4.2, but the derivation is not given. They contain substantive parameter claims about B^⊥ and A*B, so they should either be proved explicitly or clearly labeled as requiring the same, unshown case analysis.
[Examples 3.1 and 4.1] The examples assert 'It's easy to show' that the displayed matrices define an error-correcting pair. A brief verification of E.1–E.4, or at least of A*B ⊆ C^⊥, would make the examples self-contained.
[Tables 1 and 2] The tables contain formatting artifacts such as '* /AS A full /AT' and misspellings such as 'possiblities'. They should be reformatted with a clear legend explaining the notation 'full' and the meaning of the grouped rows.
[Proof of Theorem 3.1] The sentence 'd(C) = 2ℓ+1 is odd, thus n is odd' is imprecise; it is the previously established equality C = [n,2,n−2] that implies n is odd, not the parity of d(C) alone.
[Notation in Lemma 2.5] The condition I ⊊ [n] with |I| = k should be double-checked; the examples use all k-subsets of [n], so the strict subset sign may be a typo for I ⊆ [n].

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the main restrictions are derived from Product Singleton Bound and external PMDS lemmas, not from the target conclusions.

full rationale

The paper's main theorems (Theorems 3.1-3.3 and 4.1-4.3) are conditional necessary-condition results: they assume an ℓ-error-correcting pair (A,B) with A of a specified parameter case and then derive restrictions on C using Lemma 2.6, the Product Singleton Bound (Proposition 2.1), and the PMDS structure lemmas (Lemmas 2.7-2.9). These lemmas are cited from prior external work ([14], [15], [18], [19]) and are not restatements of the conclusions being proved. The enumeration of possible A-parameters is imported from the authors' earlier paper [7] via Lemmas 3.1 and 4.1, but the new theorems do not reduce to that enumeration: each case is analyzed independently and the contradictions or equalities follow from dimension and distance bounds, not from the cited enumeration's conclusion. This is self-citation, but it is not load-bearing circularity because the conditional statements would remain meaningful even if the enumeration were incomplete. Section 5 independently reproves Lemmas 2.2-2.3, further reducing reliance on the authors' prior proofs. The examples in Section 3 and 4 are explicit TGRS/GRS constructions verified through Lemma 2.5, not fitted parameters renamed as predictions. A genuine rigor gap exists: Definition 2.1 lets A and B be F_{q^m}-linear while C^⊥ is defined as an F_q-subspace, and proofs compare k(A∗B) as an F_{q^m}-dimension with k(C^⊥) as an F_q-dimension (e.g., in Theorems 3.1-3.3 and 4.1-4.3). This requires the standard convention that C^⊥ is extended to C^⊥⊗F_{q^m}; without that convention many inclusions are vacuous. However, this is a correctness/consistency issue, not a circular step: the derivations do not assume their own conclusions or rename fitted inputs as predictions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants or invented entities appear. The central claims rest on standard product Singleton and PMDS results, on the authors' previous parameter classification, and on the TGRS criterion. The least documented item is the extension-field transfer of Lemmas 2.6-2.9.

assumptions (5)

standard math Product Singleton Bound: d(A*B) <= max{1, n-k(A)-k(B)+2} (Proposition 2.1)
Invoked in Theorems 3.1-3.3 and 4.1-4.3 to bound d(A*B); cited from Randriambololona [19], not reproven.
standard math PMDS pair lemmas: if k(A)+k(B)<n, a PMDS pair has MDS components, and components are GRS with common evaluation sequence (Lemmas 2.7, 2.8)
Used to force A*B to be MDS or GRS and to derive contradictions; cited from [14,15].
standard math Distance lower bound for Schur products: if X*Y subset Z-perp, d(X-perp)>a, d(Y-perp)>b, then d(Z)>=a+b (Lemma 2.6)
Used repeatedly to bound k(B); cited from Pellikaan [18]. The field-of-definition transfer to F_{q^m} is assumed but not justified in the paper.
domain assumption He-Liao parameter classification for NMDS l-ECP: 6 cases for d=2l+1 and 10 cases for d=2l+2 (Lemma 3.1, 4.1)
The whole paper is organized around this enumeration from the authors' prior work [7]; if the enumeration is incomplete, the necessary conditions cover only a subset of possible A.
domain assumption TGRS MDS/NMDS criterion via S_{k,+}(alpha) (Lemma 2.5)
Used to certify that the constructed examples are NMDS; cited from [9]. The specific sets S_{3,+}={3,...,24} and S_{4,+}={6,...,30} are asserted without detailed proof.

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Cite this review

Pith. "Pith review of The error-correcting pair for several classes of NMDS linear codes." pith.science (2026). https://pith.science/paper/Y646RWV6

@misc{pith2026250607380,
  author       = {Pith},
  title        = {Pith review of: The error-correcting pair for several classes of NMDS linear codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y646RWV6}},
  note         = {Machine review of arXiv:2506.07380}
}

abstract

The error-correcting pair is a general algebraic decoding method for linear codes. The near maximal distance separable (NMDS) linear code is a subclass of linear codes and has applications in secret sharing scheme and communication systems due to the efficient performance, thus we focus on the error-correcting pair of NMDS linear codes. In 2023, He and Liao showed that for an NMDS linear code $\mathcal{C}$ with minimal distance $2\ell+1$ or $2\ell+2$, if $\mathcal{C}$ has an $\ell$-error-correcting pair $\left( \mathcal{A}, \mathcal{B} \right)$, then the parameters of $\mathcal{A}$ have 6 or 10 possibilities, respectively. In this manuscript, basing on Product Singleton Bound, we give several necessary conditions for that the NMDS linear code $\mathcal{C}$ with minimal distance $2\ell+1$ has an $\ell$-error-correcting pair $(\mathcal{A}, \mathcal{B})$, where the parameters of $\mathcal{A}$ is the 1st, 2nd, 4th or 5th case, then basing on twisted generalized Reed-Solomon codes, we give an example for that the parameters of $\mathcal{A}$ is the 1st case. Moreover, we also give several necessary conditions for that the NMDS linear code $\mathcal{C}$ with minimal distance $2\ell+2$ has an $\ell$-error-correcting pair $(\mathcal{A}, \mathcal{B})$, where the parameters of $\mathcal{A}$ is the 2nd, 4th, 7th or 8th case, then we give an example for that the parameters of $\mathcal{A}$ is the 1st or 2nd case, respectively.

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Works this paper leans on

25 extracted references · 25 canonical work pages

[7]

The error-correcting pair for direct sum codes

He B, Liao Q. The error-correcting pair for direct sum codes. Ad vances in Mathe- matics of Communications, 2025, 19(1): 180-201

work page 2025
[14]

A characterization of MDS codes that have an error correcting pair

M´ arquez-Corbella I, Pellikaan R. A characterization of MDS codes that have an error correcting pair. Finite Fields and Their Applications, 2016, 40: 224-2 45

work page 2016
[1]

Near-MDS codes from elliptic curves

Aguglia A, Giuzzi L, Sonnino A. Near-MDS codes from elliptic curves . Designs, Codes and Cryptography, 2021, 89: 965-972

work page 2021
[2]

Twisted Reed–Solomon codes

Beelen P, Puchinger S, Rosenkilde J. Twisted Reed–Solomon codes . IEEE Transac- tions on Information Theory, 2022, 68(5): 3047-3061

work page 2022
[3]

Decoding codes from curves and cyclic codes, Ph.D

Duursma I M. Decoding codes from curves and cyclic codes, Ph.D. thesis, Eindhoven University of Technology, 1993

work page 1993
[4]

Error-locating pairs for cyclic codes

Duursma I M, Kotter R. Error-locating pairs for cyclic codes. IE EE Transactions on Information Theory, 1994, 40(4): 1108-1121

work page 1994
[5]

On the error-correcting pair for MDS linear codes wit h even minimum distance

He B, Liao Q. On the error-correcting pair for MDS linear codes wit h even minimum distance. Finite Fields and Their Applications, 2023, 89: 102210

work page 2023
[6]

The error-correcting pair for TGRS codes

He B, Liao Q. The error-correcting pair for TGRS codes. Discret e Mathematics, 2023, 346(9): 113497

work page 2023

Show all 25 references

[8]

Constructions of MDS, Near MDS and Almost MDS Codes From Cyclic Subgroups of F∗ q2

Heng Z, Li C, Wang X. Constructions of MDS, Near MDS and Almost MDS Codes From Cyclic Subgroups of F∗ q2. IEEE Transactions on Information Theory, 2022, 68(12): 7817-7831

2022
[9]

MDS or NMDS self-dual codes from twis ted generalized Reed–Solomon codes

Huang D, Yue Q, Niu Y, et al. MDS or NMDS self-dual codes from twis ted generalized Reed–Solomon codes. Designs, Codes and Cryptography, 2021, 8 9(9): 2195-2209

2021
[10]

Fundamentals of Error-Correcting Codes

Huﬀman W C, Pless V. Fundamentals of Error-Correcting Codes . Cambridge: Cam- bridge University Press, 2003

2003
[11]

Explicit constructions of NMDS self-dual codes

Han D, Zhang H. Explicit constructions of NMDS self-dual codes . Designs, Codes and Cryptography, 2024, 92(11): 3573-3585. 19

2024
[12]

An uniﬁed description of an error locating procedure for linear codes

Kotter R. An uniﬁed description of an error locating procedure for linear codes. Proc. IAACCT, Voneshta Voda, Bulgaria, 1992

1992
[13]

Construction of MDS twisted Reed–Solomon codes an d LCD MDS codes

Liu H, Liu S. Construction of MDS twisted Reed–Solomon codes an d LCD MDS codes. Designs, Codes and Cryptography, 2021, 89(9): 2051-2 065

2021
[15]

Critical pairs for the product singleton b ound

Mirandola D, Z´ emor G. Critical pairs for the product singleton b ound. IEEE Trans- actions on Information Theory, 2015, 61(9): 4928-4937

2015
[16]

On a decoding algorithm for codes on maximal curves

Pellikaan R. On a decoding algorithm for codes on maximal curves. IEEE Transac- tions on Information Theory, 1989, 35(6): 1228-1232

1989
[17]

On decoding by error location and dependent sets of error positions

Pellikaan R. On decoding by error location and dependent sets of error positions. Discrete Mathematics, 1992, 106: 369-381

1992
[18]

On the existence of error-correcting pairs

Pellikaan R. On the existence of error-correcting pairs. Journ al of Statistical Planning and Inference, 1996, 51(2): 229-242

1996
[19]

An upper bound of Singleton type for compo nentwise products of linear codes

Randriambololona H. An upper bound of Singleton type for compo nentwise products of linear codes. IEEE Transactions on Information Theory, 2013, 59(12): 7936-7939

2013
[20]

MDS codes, NMDS codes and their secret- sharing schemes

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

2018
[21]

MDS and near-MDS codes via twisted Reed–S olomon codes

Sui J, Zhu X, Shi X. MDS and near-MDS codes via twisted Reed–S olomon codes. Designs, Codes and Cryptography, 2022, 90(8): 1937-1958

2022
[22]

Near-MDS codes from maximal arcs in PG(2 , q)

Xu L, Fan C, Han D. Near-MDS codes from maximal arcs in PG(2 , q). Finite Fields and Their Applications, 2024, 93: 102338

2024
[23]

On the error-correcting pair for a class of MDS codes

Xiao R. On the error-correcting pair for a class of MDS codes. M .S., Sichuan Normal University, 2024

2024
[24]

An Improvement for Error-Correcting Pairs of S ome Special MDS Codes

Xiao R, Liao Q. An Improvement for Error-Correcting Pairs of S ome Special MDS Codes. International Journal of Foundations of Computer Scien ce, 2024: 1-15

2024
[25]

A uniﬁed approach to construct MDS self-dua l codes via Reed- Solomon codes

Zhang A, Feng K. A uniﬁed approach to construct MDS self-dua l codes via Reed- Solomon codes. IEEE Transactions on Information Theory, 2020, 66(6): 3650-3656. 20

2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

The error-correcting pair for direct sum codes

He B, Liao Q. The error-correcting pair for direct sum codes. Ad vances in Mathe- matics of Communications, 2025, 19(1): 180-201

work page 2025

[2] [14]

A characterization of MDS codes that have an error correcting pair

M´ arquez-Corbella I, Pellikaan R. A characterization of MDS codes that have an error correcting pair. Finite Fields and Their Applications, 2016, 40: 224-2 45

work page 2016

[3] [1]

Near-MDS codes from elliptic curves

Aguglia A, Giuzzi L, Sonnino A. Near-MDS codes from elliptic curves . Designs, Codes and Cryptography, 2021, 89: 965-972

work page 2021

[4] [2]

Twisted Reed–Solomon codes

Beelen P, Puchinger S, Rosenkilde J. Twisted Reed–Solomon codes . IEEE Transac- tions on Information Theory, 2022, 68(5): 3047-3061

work page 2022

[5] [3]

Decoding codes from curves and cyclic codes, Ph.D

Duursma I M. Decoding codes from curves and cyclic codes, Ph.D. thesis, Eindhoven University of Technology, 1993

work page 1993

[6] [4]

Error-locating pairs for cyclic codes

Duursma I M, Kotter R. Error-locating pairs for cyclic codes. IE EE Transactions on Information Theory, 1994, 40(4): 1108-1121

work page 1994

[7] [5]

On the error-correcting pair for MDS linear codes wit h even minimum distance

He B, Liao Q. On the error-correcting pair for MDS linear codes wit h even minimum distance. Finite Fields and Their Applications, 2023, 89: 102210

work page 2023

[8] [6]

The error-correcting pair for TGRS codes

He B, Liao Q. The error-correcting pair for TGRS codes. Discret e Mathematics, 2023, 346(9): 113497

work page 2023