Row-Column Twisted Reed-Solomon codes

Anuj Kumar Bhagat; Harshdeep Singh; Ritumoni Sarma

REVIEW 2 major objections 4 minor 1 cited by

Row-Column Twisted Reed-Solomon codes

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Twisting rows and columns yields MDS codes inequivalent to Reed-Solomon codes

desk verdict The RCTRS construction is natural and the MDS part is solid, but the Schur-square proofs that separate the new codes from RS and CTRS codes rest on a determinant non-vanishing claim that is false for admissible parameters. read the letter →

arxiv 2509.06919 v1 pith:SDEC4EUQ submitted 2025-09-08 cs.IT math.IT

Anuj Kumar Bhagat , Harshdeep Singh , Ritumoni Sarma This is my paper

classification cs.ITmath.IT MSC 94B0594B6511T71

keywords row-columntwistedReed-SolomoncodesMDSSchursquarenon-equivalenceofcolumnfinitefieldserror-correcting

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

The paper introduces row-column twisted Reed-Solomon codes, a new family built by adding both a row twist and a column twist to the generator matrix of a Reed-Solomon code. It gives explicit conditions under which these codes are MDS and proves they exist over finite fields. By computing the dimension of their Schur squares, it argues that these MDS codes are not equivalent to Reed-Solomon codes or to column-twisted Reed-Solomon codes. If correct, this gives a genuinely new family of non-RS MDS codes, which are rare and relevant in coding theory and code-based cryptography. The paper also provides conditions for extended versions of the codes and shows they retain the same non-equivalence properties.

What carries the argument

The Schur square of a code is the span of all componentwise products of its codewords, and its dimension is invariant under Hamming-metric equivalence. The paper computes this dimension for RCTRS codes: it selects 2k or 2k+1 products of generator-matrix rows, arranges them as rows of a matrix, and reduces the determinant to a missing-row Vandermonde determinant times factors such as λbc(b−c)²η(1−η²∏α_i). Nonvanishing of these factors is what forces the Schur-square dimension to be 2k+1, separating the codes from CTRS codes (dimension 2k) and from GRS/RS codes (dimension 2k−1).

What would settle it

Work over F_19 and its quadratic extension F_{19²}, and take η with η²=2∈F_19 but η∉F_19. Choose b,c,λ and a size-9 subgroup of F*_19 as in Theorem 4.9 so that the eight chosen α_i satisfy ∏α_i = 2^{-1}; then the determinant factor 1−η²∏α_i vanishes and the claimed 2k+1 Schur-square dimension collapses. A computer search over admissible parameters decides whether such a choice exists.

Watch

Extended reading notes

Core claim

The central claim is that RCTRS codes — evaluation codes on n−1 points with an appended column f(b)−λf(c) and one generator-matrix row twisted to x^h + ηx^{k−1+t} — can be MDS and have Schur-square dimension 2k+1 in the main constructions. Since GRS/RS codes have Schur-square dimension 2k−1 and CTRS codes have dimension 2k, a code with dimension 2k+1 is inequivalent to both. The paper proves the MDS conditions via Vandermonde determinants and then proves the 2k+1 Schur-square dimension by exhibiting a linearly independent set of 2k+1 products of generator rows. This is stated for the hook h=0 case in Theorem 4.9, for h=k−1 in Theorem 5.3, and claimed for general h in Theorem 6.3, with the di

Load-bearing premise

The non-equivalence proofs need certain products, like a field parameter squared times a product of evaluation points, never to equal 1, but the stated condition that the parameter lies outside a smaller field does not prevent its square from lying inside it, so determinant nonvanishing is asserted rather than proved.

Editorial extensions

If this is right

Every RCTRS code satisfying the stated algebraic conditions is an [n,k,n−k+1] MDS code, attaining the Singleton bound.
Because Schur-square dimension is preserved by Hamming-metric equivalence, any RCTRS code with dimension 2k+1 is inequivalent to every CTRS code and to every GRS or RS code.
The subgroup-based construction yields non-RS MDS codes of length up to the order of a subgroup of F*_{q0}, including lengths (q−1)/p and (q−1)/p+1 over F_{q²}.
Extended RCTRS codes give [n+1,k,n−k+2] MDS codes with the same non-equivalence properties.
The family enlarges the known stock of non-RS MDS codes beyond twisted Reed-Solomon and column-twisted Reed-Solomon constructions.

Reading between the lines

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

The non-equivalence proofs rest on determinant factors such as 1−η²∏α_i and 1+η²(∏α_i)(Σα_i) being nonzero; the paper only assumes η lies outside a base subfield, which does not exclude η² from lying inside it. Strengthening the hypotheses to keep η² outside the base subfield would restore the determinant arguments.
Theorem 6.3, covering the general hook case, is stated without proof; the claimed 2k+1 Schur-square dimension for general h is therefore carried by an assertion rather than a written calculation.
A natural testable extension is to search, by computer, for admissible parameters where η² falls in the base subfield and the relevant determinant factor vanishes; such examples would delineate exactly where the non-equivalence claim holds.
The same row-column twisting template could be iterated with multiple twists to tune the Schur-square dimension to a prescribed value, which matters for code-based cryptography where Schur-square size is a distinguisher.

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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

The new family is sensible: glue Beelen's row twist to Liu's column twist in one generator matrix, characterize MDS via determinants, then use Schur-square dimension to separate from RS and CTRS codes. The MDS lemmas (4.1, 5.1, 6.1) are the strongest part of the paper; the determinant computations there check out and the existence theorems are standard and plausible. That part alone is worth having.

The trouble is in the Schur-square arguments that carry the headline claims. The stress-test note is correct: outside a subfield can square into it, so the assertion that η∉F_q0 forces 1−η^2∏α_i ≠ 0 fails. The same happens with the factor 1+η^2∏α_jΣα_j in Theorem 4.2. These factors are exactly what makes the exhibited 2k or 2k+1 products linearly independent. If they vanish, the rank argument collapses. This is load-bearing, not cosmetic — the non-equivalence to CTRS rests on it.

There are smaller soft spots. The paper states exact Schur-square dimensions when only lower bounds are shown. For the inequivalence conclusion a lower bound of 2k+1 would be enough, so that part is repairable by rewording. The D-set in Theorem 4.2 has duplicate entries for k=3, so the claimed 2k independent products are not defined in the stated range. And Theorem 6.3, which covers the general h case, has its proof skipped entirely; that is not acceptable in a paper claiming the result as a theorem.

The MAGMA examples are helpful but don't patch the missing arguments. The paper is honest and the construction is real; these are proof gaps, not signs of a false result. I'd send it to a serious referee, but with a clear request: fix the determinant non-vanishing, handle k=3, state lower bounds as lower bounds, and either prove or explicitly remove Theorem 6.3. The intended audience — people building non-RS MDS families or studying distinguishers for code-based crypto — should read the MDS portions now and treat the equivalence claims as provisional until repaired.

Referee Report

2 major / 4 minor

Summary. The paper introduces row-column twisted Reed-Solomon (RCTRS) codes, obtained by combining a row twist (as in twisted RS codes) with a column twist (as in column-twisted RS codes). It provides generator matrices, derives MDS conditions for three families (h=0, h=k-1, and general h), gives subfield-based constructions satisfying those conditions, and claims, via Schur-square dimension computations, that the resulting MDS codes are neither equivalent to Reed-Solomon codes nor to column-twisted Reed-Solomon codes. Theorems 4.9, 5.3, and 6.3 are the main non-equivalence results.

Significance. If the claims are correct, the paper adds a new family of non-RS MDS codes and distinguishes it from the recently introduced CTRS family. The paper has clear strengths: explicit generator matrices, explicit MDS criteria in Lemmas 4.1, 5.1 and 6.1, several worked examples with MAGMA verification, and a concrete subfield construction giving lengths up to (q-1)/p and (q-1)/p+1. The determinant computations for the MDS conditions appear standard and sound. However, the Schur-square arguments contain a field-theoretic gap in the non-vanishing of the displayed determinants, and one of the central non-equivalence theorems is left without proof. These issues directly affect the paper's main novelty claim and require substantial repair.

major comments (2)

[Theorems 4.2 and 4.9 (§4)] The claimed non-vanishing of the displayed determinants is not implied by the hypothesis η∉F_q0. In Theorem 4.2 the rank criterion is λbc(b−c)^2(1+η²∏_{j=1}^{2k−1}α_j ∑_{j=1}^{2k−1}α_j)≠0, and in Theorem 4.9 it is λbc(b−c)^2η(1−η²∏_{i=1}^{2k}α_i)≠0. The proofs argue that because η∉F_q0 the parentheses are non-zero. This is false over finite fields: an element outside a field can have its square inside the field. Concretely, in Theorem 4.2 take q0=F7, q1=F_{7^3}, q=F_{7^6}, α_1,...,α_5=1,2,3,4,5, b,c∈F7^* distinct, λ∈F_{7^3}^*\F7, and η∈F_q with η²=6. Then η∉F_{q1}^* and all hypotheses hold, but ∏α_j∑α_j=1 and 1+η²·1=0, so the 2k×2k submatrix D′ is singular and the displayed set D is not proved independent. The MDS conditions in Lemma 4.1 do not exclude such parameters. The same defect invalidates the proof of Theorem 4.9. A repair must add an explicit condition such as η²∉{...}, or give
[Theorem 6.3 (§6)] The proof of Theorem 6.3 is explicitly skipped: 'We skip the proof.' This is a central theorem claiming that general RCTRS_{h,1} codes are not equivalent to CTRS codes. The preceding Theorem 6.2 also asserts, without proof, that the sets D and D1 are bases of the Schur square, saying only 'it is not difficult to prove.' Given that the analogous independence claims in Theorems 4.2 and 4.9 contain a subtle non-vanishing error, this omitted proof cannot be treated as routine. The authors need to provide complete arguments for both the dimension computation and the non-equivalence conclusion.

minor comments (4)

[Theorems 4.2, 4.9, 5.3] The text concludes 'dim C^{⋆2}=2k+1' (or '=2k') immediately after proving that a displayed set is linearly independent. This establishes only a lower bound, not equality. For the non-equivalence-to-CTRS claim the lower bound is sufficient, and for the non-RS claim a lower bound of 2k is also sufficient, so this does not affect the central conclusion; nevertheless the statements as written overclaim.
[Definition 3.2] In the second displayed equation of Definition 3.2, the text says 'an extended CTRS code is defined as' but it is defining the extended RCTRS code. This is a typo that should be corrected.
[Example 4.8] Example 4.8 uses η=4∈F_{17}^*, while Theorem 4.4 requires η∈F_q\F_{q0}^*. The example is presented as evidence that one may also obtain non-RS MDS codes for η∈F_{q0}^*, but it lies outside the theorem's hypotheses. Please clarify whether this is only a computational example or a new existence statement with its own proof.
[Theorem 4.2] The hypothesis is written as η∈F_q\F_{q1}^*, but the proof repeatedly derives a contradiction from 'η∈F_{q1}^*'. The case η=0 is handled implicitly, but the argument would be clearer if the intended hypothesis were stated as η∈F_q\F_{q1} or if the zero case were explicitly separated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained.

full rationale

The paper introduces RCTRS codes by explicit generator-matrix definitions and proves MDS conditions and Schur-square dimensions from first-principles determinant computations. The comparison against Reed-Solomon and column-twisted Reed-Solomon codes uses external benchmarks (Lemma 2.5 and the Schur-square dimension result of [19]) rather than results derived within the paper, so there is no self-citation load-bearing step and no fitted parameter renamed as a prediction. The flagged issue that η outside a subfield does not imply the displayed determinant factors are nonzero is a mathematical correctness gap in the proof of the non-equivalence theorem, not a circularity: the claimed Schur-square dimension is being derived, not assumed, and if the determinant can vanish for admissible parameters then the proof would fail without making the argument circular. Accordingly, no step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction uses only standard finite-field algebra. No numerical constants are fitted; η, λ, and the evaluation points are free construction variables chosen to avoid finitely many bad subsets, not parameters fit to data. The main external input is [19]'s Schur square dimension for CTRS codes, which is a prior result, not an assumption of the present conclusion. The gap in the non-vanishing argument is a correctness issue, not circularity.

assumptions (4)

standard math A linear code is MDS iff every k columns of a generator matrix are linearly independent.
Invoked as Lemma 2.8 and used in the proofs of Lemmas 4.1, 5.1, 6.1.
standard math Equivalent codes have equivalent Schur squares, so Schur square dimension is an equivalence invariant.
Remark 2.7, the basis for all non-equivalence conclusions.
domain assumption The [n,k] CTRS and extended CTRS codes of Liu et al. have Schur square dimension 2k.
External result cited from [19]; used to conclude RCTRS codes with dimension at least 2k+1 are not equivalent to CTRS.
domain assumption The evaluation points α_i are distinct and the field/subfield/group choices make the relevant products and sums lie in the stated subfields.
Needed for Vandermonde determinants to be nonzero and for the subfield arguments in Theorems 4.2, 4.4, 5.2, 6.2.

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Cite this review

Pith. "Pith review of Row-Column Twisted Reed-Solomon codes." pith.science (2026). https://pith.science/paper/SDEC4EUQ

@misc{pith2026250906919,
  author       = {Pith},
  title        = {Pith review of: Row-Column Twisted Reed-Solomon codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDEC4EUQ}},
  note         = {Machine review of arXiv:2509.06919}
}

read the original abstract

In this article, we present a new class of codes known as row-column twisted Reed-Solomon codes (abbreviated as RCTRS), motivated by the works of \cite{beelen2017twisted} and \cite{liu2025column}. We explicitly provide conditions for such codes to be MDS and also ensure their existence. By determining the dimensions of their Schur squares, we prove that these MDS codes are not equivalent to Reed-Solomon codes, thus presenting a new family of non-RS MDS codes. Additionally, we prove that these MDS codes are also not equivalent to column twisted Reed-Solomon codes described in \cite{liu2025column}, showing the novelty of our construction.

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Forward citations

Cited by 1 Pith paper

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

The Hermitian Hull Dimensions for a Class of (L,P)-Twisted Generalized Reed-Solomon Codes
cs.IT 2026-07 accept novelty 6.0 of 10

Hermitian hull dimensions of the codes C_{q+j}(α) are fully determined in three cases on i and q+1, producing EAQECC parameters [[i(q-1), q-1+m, d, (i-2)(q-1)+m]]_q for m=0,1,2,3.

Reference graph

Works this paper leans on

31 extracted references · 26 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Abdukhalikov, C

K. Abdukhalikov, C. Ding, and G. K. Verma. Some constructions of non-generalized Reed-Solomon MDS Codes.arXiv preprint arXiv:2506.04080, 2025

arXiv 2025

[2] [2]

T. L. Alderson. On the weights of general MDS codes.IEEE Transactions on Information Theory, 66(9):5414–5418, 2020

work page 2020

[3] [3]

S. Ball. On large subsets of a finite vector space in which every subset of basis size is a basis.J. Eur. Math. Soc, 14(3):733–748, 2012

work page 2012

[4] [4]

Bartoli, M

D. Bartoli, M. Giulietti, and I. Platoni. On the covering radius of MDS codes.IEEE Transactions on Information Theory, 61(2):801–811, 2014

work page 2014

[5] [5]

Beelen, M

P. Beelen, M. Bossert, S. Puchinger, and J. Rosenkilde. Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography. In2018 IEEE International Symposium on Information Theory (ISIT), pages 946–950, 2018

work page 2018

[6] [6]

Beelen, S

P. Beelen, S. Puchinger, and J. R. n´ e Nielsen. Twisted Reed-Solomon codes. In2017 IEEE International Symposium on Information Theory (ISIT), pages 336–340. IEEE, 2017

work page 2017

[7] [7]

Beelen, S

P. Beelen, S. Puchinger, and J. Rosenkilde. Twisted Reed–Solomon codes.IEEE trans- actions on information theory, 68(5):3047–3061, 2022

work page 2022

[8] [8]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user lan- guage.J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993)

work page 1997