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REVIEW 3 major objections 5 minor 35 references

On subcodes of the generalized Reed-Solomon codes

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper pinpoints exactly when one-row-deleted subcodes of generalized Reed-Solomon codes are self-dual or near-MDS.

desk verdict Core characterizations are sound but sign errors in Lemmas 2.3/2.4 and a false Example 3.1 need fixing before this is publishable. read the letter →

arxiv 2507.04689 v1 pith:LV7SHTPP submitted 2025-07-07 cs.IT math.IT

Yu Ning This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0594B2711T71
keywords generalizedReed-Solomoncodessubself-dualnear-MDStwistedGRSdualelementarysymmetricpolynomialsfinitefields
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 aims to pin down, for every choice of which row is deleted, whether the resulting subcode is self-dual and whether it is near-MDS, one step short of MDS. The answer it proposes is a three-part rule: deleting a row of interior degree $2\le r\le k-2$ can never give a self-dual code, whereas deleting the first or last allowed degree gives self-duality exactly when a power-sum vanishes and the associated weights $u_i$ or $u_i s_{n-2}(a_i)$ are all squares or all non-squares. For near-MDS behavior the paper gives a uniform criterion: the code is NMDS or MDS precisely when elementary symmetric sums of degrees $r-1$ and $r$ do not both vanish on any $(k-1)$-subset of evaluation points, and the last allowed deletion is always NMDS or MDS. The paper also writes the dual codes explicitly for $r=1,2$ and $k-1$, showing that they are again one-row-deleted GRS subcodes or twisted GRS codes. Since self-dual and near-MDS codes are standard building blocks in coding-theoretic cryptography and designs, exact characterizations of this kind turn construction problems into finite-field arithmetic.

What carries the argument

The load-bearing object is the generator matrix $G_{k,r}(a,v)=V_{J(k,r)}(a)D(v)$ with row set $J(k,r)=[0,k]\setminus\{k-r\}$, so the code consists of evaluations of polynomials of degree at most $k$ with the monomial of degree $k-r$ removed. Self-duality is checked through the Gram condition $G_{k,r}G_{k,r}^{\mathsf T}=0$; the sumset identity $J(k,r)+J(k,r)=J(2k,1)$ for $r=1$, $J(2k,2k-1)$ for $r=k-1$, and $[0,2k]$ for $2\le r\le k-2$ reduces that matrix equation to the vanishing of weighted power sums. The paper's two power-sum lemmas, Lemma 2.3 for weights $u_i$ and Lemma 2.4 for weights $u_i s_{n-2}(a_i)$, then identify the one-dimensional kernel exactly, forcing the factor vector $v^2$ to be a scalar multiple of $u$ or of $(u_i s_{n-2}(a_i))$, which is precisely the square/non-square condition. The NMDS analysis instead uses the column-rank test: any $k-1$ columns of $G_{k,r}$ have rank $k-1$ exactly when $s_{k-1,r-1}(A)$ and $s_{k-1,r}(A)$ do not both vanish for a $(k-1)$-subset $A$ of evaluation points, and for the extended code the analogous condition is imposed on subsets of size $k-2$.

What would settle it

Compute Lemma 2.4 directly over $\mathbb{F}_{11}$ with $n=5$, taking $a=(1,2,3,4,5)$: the lemma predicts $\sum_{i=1}^5 a_i u_i s_3(a_i)=-1$, i.e. $10$ in $\mathbb{F}_{11}$, so any other value refutes the lemma and, with it, Theorems 3.5, 3.7 and 5.2.

Watch

Extended reading notes

Core claim

The paper studies the $[n,k]$ codes $\mathrm{GRS}_{k,r}(a,v)$ obtained from an $[n,k+1]$ generalized Reed-Solomon code by deleting the generator row of degree $k-r$, with $1\le r\le k-1$. It proves that for $2\le r\le k-2$ neither $\mathrm{GRS}_{k,r}(a,v)$ nor its extended analogue can ever be self-dual, for any evaluation points $a$ and nonzero factors $v$. For $r=1$, self-dual factors exist exactly when the sum $t_1=\sum_i a_i$ is zero and the weights $u_i=\prod_{j\ne i}(a_i-a_j)^{-1}$ are simultaneously squares or non-squares in $\mathbb{F}_q$; for $r=k-1$, the same characterization holds with $t_{n-1}=0$ and with $u_i$ replaced by $u_i s_{n-2}(a_i)$, where $a_i=a\setminus\{a_i\}$. On the NMDS side, for $r\ge 2$ the code is NMDS or MDS exactly when no $(k-1)$-subset $A$ of evaluation points has both $s_{k-1,r-1}(A)=0$ and $s_{k-1,r}(A)=0$, and the case $r=k-1$ is always NMDS or MDS. Finally, for $r=1$ with zero evaluation sum the dual is again a one-row-deleted GRS subcode, for $r=k-1$ with $t_{n-1}=0$ the dual is $\mathrm{GRS}_{n-k,n-k-1}(a,(u_i s_{n-2}(a_i)))$, and for $r=2$ the dual is an explicitly described evaluation space that is a twisted GRS code in most of the five possible cases.

Load-bearing premise

The $r=k-1$ results rest on Lemma 2.4, stated without proof, which fixes the weighted power sums $\sum_{i=1}^n a_i^{\ell} u_i s_{n-2}(a_i)$ for $\ell\in\{0,1,2,\dots,n-1,n,n+1\}$; if that identity is wrong, the self-dual characterization and the dual-code formulas for the top deleted degree fail.

Editorial extensions

If this is right

  • Every code $\mathrm{GRS}_{k,r}(a,v)$ with $2\le r\le k-2$ and $n=2k$ is non-self-dual, no matter how the factors are chosen, and the same holds for the extended codes when $n+1=2k$.
  • When $n=2k$ and $t_1=0$, self-dual factors exist iff the $u_i$ are simultaneous squares or non-squares, so every known construction of self-dual GRS codes yields a self-dual $\mathrm{GRS}_{k,1}(b,w)$ after shifting the evaluation points so that their sum is zero.
  • For $r=k-1$ and $n=2k$, self-dual $\mathrm{GRS}_{k,k-1}(a,v)$ codes exist exactly when $t_{n-1}=0$ and the $u_is_{n-2}(a_i)$ are simultaneous squares or non-squares; a concrete family is realized on cyclic subgroups of $\mathbb{F}_q^*$ when $n$ divides $q-1$.
  • For $2\le r\le k-1$, $\mathrm{GRS}_{k,r}(a)$ is NMDS or MDS exactly when no $(k-1)$-subset $A$ has $s_{k-1,r-1}(A)=s_{k-1,r}(A)=0$, and the case $r=k-1$ is always NMDS or MDS.
  • For $r=1$ with $t_1=0$, the dual is again a one-row-deleted GRS subcode; for $r=2$, the dual has five explicit algebraic forms depending on the power sums $h_1,h_2$, with twisted GRS codes appearing in most cases.

Reading between the lines

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

  • If the square/non-square criteria are right, checking self-duality of a candidate factor vector reduces to testing whether a single vector of weights lies in one square class, so random search over $\mathbb{F}_q^n$ can be replaced by a deterministic field-arithmetic check.
  • The never-self-dual middle regime is a rigidity statement: restoring self-duality there would require modifying the row structure, for example by twisting or extending the deleted row, since the obstruction is precisely that the row-sum set covers all powers $0$ through $2k$.
  • A full proof of Lemma 2.4, which is currently stated without proof, is the one missing link in the top-degree results; a detailed coefficient comparison following the proof of Lemma 2.3 would either complete the chain or expose a hidden field-characteristic failure for small $q$.
  • The five-case dual classification for $r=2$ suggests an iterative pattern: applying the same column-swap argument to higher $r$ would likely produce duals that are evaluation spaces with several extra polynomials, interpolating between ordinary GRS subcodes and multi-twisted TGRS codes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the [n,k]_q subcodes GRS_{k,r}(a,v) of generalized Reed-Solomon codes obtained by deleting the row of degree k-r from the generator matrix of an [n,k+1]_q GRS code, together with the analogous extended codes GRS_{k,r}(a,v,∞). The main results are: self-duality is impossible for 2≤r≤k−2 (Theorem 3.1); self-duality for r=1 is characterized by t_1=0 and the u_i having a common quadratic character (Theorem 3.3); self-duality for r=k−1 is characterized by t_{n−1}=0 and the u_i s_{n−2}(a_i) having a common quadratic character (Theorem 3.5); GRS_{k,r}(a) is NMDS or MDS exactly under the stated non-vanishing conditions on elementary symmetric polynomials (Theorem 4.2 and Corollary 4.3); and the dual codes for r=1,2 are described as GRS subcodes or twisted GRS codes (Theorems 5.1, 5.2, 5.3, 5.4). The paper also gives explicit constructions of self-dual codes, including Example 3.1.

Significance. If the main theorems are correct, the paper gives useful, checkable algebraic characterizations rather than heuristic constructions: self-duality is reduced to power sums and quadratic characters, and NMDS status is reduced to non-vanishing of elementary symmetric polynomials. The dual-code descriptions, especially the appearance of twisted GRS codes, connect this class of subcodes to a currently active area. The arguments are parameter-free and appear to extend the r=1 results of Han-Zhang and Li-Sun-Zhu in a natural way. No fitted parameters or post-hoc selections are involved. The main barrier is not the central derivation but a false construction in Example 3.1 and some missing foundational proof details.

major comments (3)
  1. [Section 3.2, Example 3.1] Example 3.1 is invalid as stated. For q=17 and n=16, the evaluation points are F_17^*, whose elements are not all squares. The example's own computation gives w_i = u_i s_{n-2}(a_i) = 1/(16 a_i) = -1/a_i, and since -1 is a square in F_17, the w_i are squares exactly for square a_i and non-squares for non-square a_i. Thus the w_i are not simultaneously squares or non-squares, and Theorem 3.5 implies that GRS_{8,7}(a,v) is not self-dual for any non-zero v. The family can be repaired by requiring n | (q−1)/2 instead of n | (q−1); with that change the argument goes through for even n.
  2. [Section 2.4, Lemmas 2.3 and 2.4] There is an inconsistency in the normalization of u_i. The displayed definition u_i = ∏_{j≠i}(a_j−a_i)^{-1} makes Lemma 2.3 false for even n: for n=2 and a=(0,1), the l=n−1 sum equals −1, not 1. Equation (18) and Example 3.1 instead use the standard convention u_i = ∏_{j≠i}(a_i−a_j)^{-1}. The paper should adopt one convention throughout; under the standard convention the stated formulas of Lemmas 2.3 and 2.4 are correct.
  3. [Sections 2.4, 3.2, and 5] Lemma 2.4 is stated with proof omitted, but it is load-bearing for Theorems 3.5, 3.7, and 5.2; it is not a one-line consequence of Lemma 2.3 and a proof should be included. More generally, the omitted proofs of Theorem 3.2, Corollary 4.7, Theorem 5.2, and Theorem 5.4 should be supplied or replaced by precise references, since some of these are central to the dual-code claims.
minor comments (5)
  1. [Theorem 3.5, Eq. (23)] The matrix M should be the n×n matrix with rows indexed by [0,2k]\{1}; the notation 'a^0, a^2, ..., a^{2k}' is ambiguous and the later notation 'f_0, f_2, ..., f_{2k}' should make clear that the index 1 is omitted while all other degrees are present.
  2. [Example 3.1] The displayed quotient-rule expression for f'_{a_i}(x) should have f_a(x), not f_{a_i}(x), in the numerator; the printed version is incorrect as a derivative of f_a(x)/(x−a_i).
  3. [Corollary 4.3] The statement 'GRS_{k,k−1}(a) is always NMDS or NMDS' should read 'NMDS or MDS'.
  4. [Theorem 5.3] In case (3), 'h2_1' should be 'h1^2'; in case (5), the formula needs parentheses, e.g., g(x) = x^{n-k-1} − (h_1 x^{n-k} − x^{n-k+1})/(h_1^2 − h_2).
  5. [Theorem 3.7] In the converse part, the notation s_{n-1,n-2}(a_i) should be s_{n-2}(a_i), and the sentence claiming that v_1,...,v_n are all squares should be phrased as the existence of square roots under the theorem's hypothesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central equivalences are derived from re-proved Lagrange interpolation identities and direct rank arguments; the only self-citation is an independent published characterization used for constructions.

full rationale

The derivation chain is self-contained for the main results. The self-duality criteria (Theorems 3.3 and 3.5) are proved by setting GG^T = 0, reducing to a Vandermonde-kernel condition, and identifying the kernel vector via Lemma 2.3, which is fully re-proved by Lagrange interpolation, or via Lemma 2.4, which is an algebraic power-sum identity. The relevant vectors u_i and u_i s_{n-2}(a_i) arise from these identities rather than being assumed from the conclusion. The r in [2,k-2] non-self-duality result uses only the set identity J(k,r)+J(k,r)=[0,2k] and the invertibility of a Vandermonde submatrix. The NMDS characterizations follow from rank computations on column submatrices (Lemmas 4.1 and 4.5), and the dual-code formulas are checked by direct dot products. The one self-citation, Theorem 2.1 from the author's prior work [24], is a previously published characterization of self-dual GRS codes; it is used only in Corollary 3.4 to transfer known constructions and is not needed for the paper's main equivalence proofs. No fitted parameters are renamed as predictions, and no result is defined in terms of its target. Two weaknesses noted by a skeptical reader, namely Lemma 2.4 being left unproved and Example 3.1 asserting that a subgroup of order n dividing q-1 consists of squares, are correctness or completeness risks rather than circularity, because neither assumes the theorem being proved.

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

The paper's results rest on standard finite-field facts (Vandermonde matrices, power sums, Singleton bound) plus two internal ingredients: the unproved power-sum identity Lemma 2.4 and the elementary set-sum identity Eq. (21). No free parameters or invented entities are introduced.

assumptions (5)
  • standard math Singleton bound and definitions of MDS, AMDS, and NMDS
    Used throughout to interpret minimum distances and to conclude that GRS_{k,r} is NMDS or MDS when d_perp >= k (Sections 1 and 4).
  • standard math Power-sum identity Lemma 2.3: sum_i u_i a_i^l = 0 for l <= n-2, = 1 for l = n-1, = t_1 for l = n
    Proved via Lagrange interpolation in Section 2.4; it is the engine for the self-dual and dual-code computations.
  • domain assumption Lemma 2.4 power-sum identity with weights u_i s_{n-2}(a_i)
    Stated without proof; used centrally in Theorems 3.5, 3.7 and 5.2 to identify kernel vectors and power sums t_{n-1}, t_n.
  • standard math Set-sum identity Eq. (21): J(k,r)+J(k,r) = [0,2k] for 2 <= r <= k-2
    Elementary arithmetic fact left as 'easy to check'; it makes GG^T = 0 equivalent to vanishing of all power sums in Theorem 3.1.
  • standard math Theorem 2.1 on self-dual GRS codes from [24,33]
    External characterization used in Corollary 3.4 to transfer self-duality from GRS_k(a,v) to GRS_{k,1}(b,w).

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Pith. "Pith review of On subcodes of the generalized Reed-Solomon codes." pith.science (2026). https://pith.science/paper/LV7SHTPP

@misc{pith2026250704689,
  author       = {Pith},
  title        = {Pith review of: On subcodes of the generalized Reed-Solomon codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LV7SHTPP}},
  note         = {Machine review of arXiv:2507.04689}
}
abstract

In this paper, we study a class of subcodes of codimension $1$ in the $[n,k+1]_q$ generalized Reed-Solomon (GRS) codes, whose generator matrix is derived by removing the row of degree $k-r$ from the generator matrix of the $[n,k+1]_q$ GRS codes, where $1 \le r \le k-1$. We show equivalent characterizations for this class of subcodes of the GRS codes being self-dual or near-MDS, which extends the results for $r=1$ in the literature. Along with these characterizations, families of self-dual near-MDS subcodes of the GRS codes are also proposed. Finally, for $r = 1,2$, the dual codes of the subcodes of the GRS codes are found out. In some cases, the subcodes of the GRS codes can be closed under taking dual codes. In other cases, the dual codes turn out to be the twisted GRS codes.

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Reference graph

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