On Euclidean Hulls of MDS Codes

Jinquan Luo; Meiqing Liu; Xiaolei Fang

REVIEW 2 major objections 3 minor 34 references

On Euclidean Hulls of MDS Codes

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read From any self-orthogonal generalized Reed-Solomon code, this paper builds maximum distance separable codes with every prescribed Euclidean hull dimension.

desk verdict Theorems 1 and 2 are correct and genuinely general, but Corollary 3.2(ii) has a proof gap: the extended GRS code of a self-dual GRS seed is not self-orthogonal in general, so the stated derivation from Theorem 2 does not work. read the letter →

arxiv 1908.01173 v3 pith:T7RNWVMM submitted 2019-08-03 cs.IT math.IT

Xiaolei Fang , Meiqing Liu , Jinquan Luo This is my paper · ORCID

classification cs.ITmath.IT MSC 94B0594B2711T71

keywords MDScodesEuclideanhullgeneralizedReed-Solomonself-orthogonalself-dualdimensionentanglement-assistedquantumerror-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

This paper establishes a general mechanism for building maximum distance separable (MDS) codes with a prescribed Euclidean hull dimension (the dimension of the code's intersection with its dual). Starting from a self-orthogonal generalized Reed-Solomon (GRS) code of dimension $m$, it constructs, for any $k \leq m$ and any $l$ between $0$ and $k$, an MDS code of the same length and dimension $k$ whose hull has dimension exactly $l$; the extended-GRS analogue adds one coordinate. The proof rescales the first $k-l$ entries of the code's multiplier vector by a scalar whose square is not $1$, which forces hull codewords to vanish at those positions. Because codes with assigned hull dimensions are the classical ingredient for entanglement-assisted quantum error-correcting codes (EAQECCs), the result makes those quantum codes flexible in their parameters, and it unifies earlier constructions of MDS codes with small or zero hulls.

What carries the argument

The load-bearing object is the multiplier vector $\mathbf{v}$ of a GRS code together with the numbers $u_i = \prod_{j \neq i}(a_i - a_j)^{-1}$. Lemma 2.4 characterizes self-orthogonality: $\mathrm{GRS}_m(\mathbf{a},\mathbf{v})$ is Euclidean self-orthogonal if and only if $v_i^2 = \lambda(a_i)u_i$ for every $i$, with $\lambda$ a polynomial of degree at most $n-2m$; Lemma 2.5 gives the extended analogue with $\lambda(x) = \lambda_0 + \cdots + \lambda_{n-2m}x^{n-2m} - x^{n-2m+1}$. Given such a code, the construction scales the first $s$ entries of $\mathbf{v}$ by $\alpha$ with $\alpha^2 \neq 1$. Comparing the first $s$ positions with the remaining $n-s$ positions in the hull equations (Lemmas 2.1 and 2.2) yields $(\alpha^2 - 1)\lambda(a_i)u_i f(a_i) = 0$, so $f(a_i) = 0$ for $i \leq s$. Conversely every polynomial divisible by $\prod_{i=1}^s (x-a_i)$ gives a hull codeword, so the hull has exactly dimension $k-s$.

What would settle it

Over $\mathbb{F}_9$, take a self-orthogonal $\mathrm{GRS}_2(\mathbf{a},\mathbf{v})$ satisfying the condition of Lemma 2.4, set $s=1$, pick $\alpha$ with $\alpha^2 \neq 1$, and form $\mathbf{v}'$ by scaling the first coordinate; computing the hull dimensions of $\mathrm{GRS}_1(\mathbf{a},\mathbf{v}')$ and $\mathrm{GRS}_2(\mathbf{a},\mathbf{v}')$ should give $0$ and $1$ respectively. Alternatively, one can check a square-residue claim in Example 4.1 directly: for the constructed set $A$, test whether $\prod_{z \neq i}(a_i - a_z)$ is a square in $\mathbb{F}_q$ for every $i$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: assume $1 \leq m \leq \lfloor n/2 \rfloor$ and $q > 3$, and suppose the GRS code $\mathrm{GRS}_m(\mathbf{a},\mathbf{v})$ is Euclidean self-orthogonal (contained in its Euclidean dual). Then for every $0 \leq l \leq k \leq m$ there exists a $q$-ary $[n,k]$ MDS code $C$ with $\dim \mathrm{Hull}(C) = l$. Theorem 2 gives the extended-GRS version: under $n < q$, a self-orthogonal extended GRS code of dimension $m$ yields $[n+1,k]$ MDS codes with any hull dimension $l \leq k \leq m$. The construction is explicit: set $s = k-l$, pick $\alpha \in \mathbb{F}_q^*$ with $\alpha^2 \neq 1$, and form $\mathbf{v}'$ by multiplying the first $s$ coordinates of $\mathbf{v}$ by $\alpha$ (for extended codes, also by the factor $(x-b)^{m-k}$ evaluated at each $a_i$ with $b \notin \{a_1,\dots,a_n\}$). The hull of $C = \mathrm{GRS}_k(\mathbf{a},\mathbf{v}')$ consists exactly of codewords whose defining polynomial $f$ has $a_1,\dots,a_s$ as roots, so its dimension is $k-s = l$.

Load-bearing premise

The construction assumes that a self-orthogonal (extended) GRS code of dimension $m$ with the multiplier form $v_i^2 = \lambda(a_i)u_i$ actually exists; the infinite families in the examples depend on square-residue claims from the authors' earlier paper that are not proved here.

Editorial extensions

If this is right

Every known self-orthogonal or self-dual (extended) GRS code automatically yields MDS codes with all hull dimensions from $0$ up to $k$, removing the need for separate constructions for each hull size.
The resulting codes with hull dimension $l$ translate into entanglement-assisted quantum error-correcting codes whose number of required entangled states is tied to $l$, so the construction makes EAQECC parameters flexible.
Corollaries 3.1 and 3.2 recover earlier MDS hull constructions as special cases, so the mechanism subsumes previously known results.
Example 4.4 supplies a family where no self-dual MDS code can exist ($q \equiv 3 \pmod{4}$, $n \equiv 2 \pmod{4}$), showing the self-orthogonal assumption reaches cases the self-dual route cannot.

Reading between the lines

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

The scaling trick seems to be independent of the GRS structure beyond the hull equation, so a similar 'rescale the generator to pin the hull' mechanism may apply to any code whose hull can be characterized by polynomial vanishing; the paper leaves this open.
The same argument with $\alpha^2 \neq 1$ replaced by a $q$-th power condition could plausibly construct Hermitian hulls of prescribed dimensions, an extension the paper does not make.
The $q=3$ examples in Remark 3.3 suggest the condition $q > 3$ is not an artifact of the proof but a genuine threshold, since a 3-ary $[4,2]$ code cannot have hull dimension $1$.
One direct check: for a small field, pick any self-orthogonal $\mathrm{GRS}_2$ code, choose $\alpha = 2$ say, and compute the hull of $\mathrm{GRS}_1$ and $\mathrm{GRS}_2$ with the scaled multiplier; the dimensions should match the theorem exactly.

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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 main theorems check out. The mechanism is simple and clean: start with any self-orthogonal (extended) GRS code, scale a prefix of the multiplier vector by a constant α with α² ≠ 1, and the hull of the resulting MDS code has exactly the desired dimension. Prior constructions required self-dual seeds; this paper opens the door to any self-orthogonal seed, which is a real generalization. The degree counting in Lemmas 2.4 and 2.5 is correct, and the polynomial-identity arguments in both theorems are sound. The examples lean on the authors' earlier preprint [12] for the seed codes, so they are conditional, but the theorems themselves are self-contained.

The soft spot is Corollary 3.2(ii). The paper says it follows directly from Theorem 2 by taking a self-dual GRS_{n/2}(a,v). That does not work. Self-duality of the nonextended code gives v_i² = λ0 u_i with λ0 constant (Corollary 2.1). To invoke Theorem 2, the extended code GRS_{n/2}(a,v,∞) would need to be self-orthogonal, and Lemma 2.5 forces v_i² = (λ0 – a_i) u_i, which is not constant. So the seed does not satisfy the hypothesis. The same gap propagates to Example 4.1(ii), which explicitly uses Corollary 3.2(ii). The corollary statement appears to be true—the stress-test note sketches a direct hull computation with π(x) = (x – b)^{m-k} and s = k–l–1 that avoids the extended self-orthogonality condition—but as written the proof is incomplete. This is a genuine, localized flaw, not a crack in the main edifice.

Who this is for: people working on MDS codes with prescribed hull dimensions, and those building EAQECCs from hulls. The significance is subfield-level, but the mechanism is likely to be reused.

Recommendation: send to peer review. The main theorems are correct and the gap in Corollary 3.2(ii) is fixable; a competent referee will flag it and the authors can repair the proof without changing the paper's core contribution.

Referee Report

2 major / 3 minor

Summary. The paper proposes a general mechanism for constructing q-ary MDS codes with a prescribed dimension l of the Euclidean hull. The input is a self-orthogonal GRS code GRS_m(a,v) (Theorem 1) or extended GRS code GRS_m(a,v,∞) (Theorem 2) of dimension m; from it the authors construct, for any 0 ≤ l ≤ k ≤ m, an [n,k] (or [n+1,k]) MDS code with hull dimension exactly l. The construction multiplies an initial segment of the entries of v by a scalar α with α^2≠1 and, in the extended case, factors in a polynomial π(x) of degree m-k. A hull-vector argument shows that the hull consists precisely of those codewords whose associated polynomial vanishes at the first s=k-l coordinates. Several corollaries and concrete families (q=r^2, q=p^{2s}, q≡3 mod 4) are derived from known self-dual and almost self-dual GRS seeds.

Significance. The proposed mechanism is clean and, conditional on the existence of suitable self-orthogonal seeds, delivers MDS codes with arbitrary hull dimensions, subsuming several earlier constructions (Remarks 3.1 and 3.2). The proofs of Theorems 1 and 2 are complete: the degree bounds in Lemmas 2.4 and 2.5 are correct, and the polynomial-identity arguments that identify the hull are rigorous. The paper makes the role of the seed codes transparent and frames a natural open problem. However, the proof of Corollary 3.2(ii) has a gap, and the examples depend on the authors' preprint [12] for seed existence, so the paper as submitted is not yet ready for acceptance.

major comments (2)

[Section 3, Corollary 3.2(ii) and Section 4, Example 4.1(ii)] The claimed direct derivation of Corollary 3.2(ii) from Theorem 2 is invalid. Self-duality of GRS_{n/2}(a,v) gives v_i^2 = λ_0 u_i with λ_0 constant by Corollary 2.1, whereas Theorem 2 requires the extended code GRS_{n/2}(a,v,∞) to be self-orthogonal; by Lemma 2.5 that would require v_i^2 = (λ_0 - a_i)u_i, a condition not implied by the seed. The same gap appears in the proof of Example 4.1(ii), which invokes Theorem 2 after the same self-dual GRS seed. The corollary may be salvageable by a separate polynomial-identity argument, but the paper does not provide it.
[Section 4, Examples 4.1–4.3] The constructions of the self-dual or almost self-dual seeds rely on square-residue statements taken from the authors' preprint [12] (e.g., 'by [12]' in the proofs of Examples 4.1 and 4.2). None of these statements is reproduced or proved in the present manuscript. If [12] is not available to the reader, the examples are not verifiable. The authors should include a proof or a precise statement of the needed lemmas, or explicitly label the examples as conditional on [12].

minor comments (3)

[Abstract] The abstract contains typographical artifacts: 'MD S codes' and 'g eneralized' should be 'MDS codes' and 'generalized'.
[Remark 3.3(iii)] The assertion that no 3-ary [4,2,3] code with hull dimension 1 exists is not accompanied by a calculation or a reference; please provide the verification.
[Section 2, proof of Lemma 2.4] The phrase 'linear independent' should be 'linearly independent'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems prove hull dimensions from a stated self-orthogonality assumption, and the cited self-dual seed results are external instantiations, not fitted inputs.

full rationale

The derivation chain is self-contained and conditional. Theorem 1 assumes GRS_m(a,v) is self-orthogonal and, via Lemma 2.4, represents v_i^2 = lambda(a_i)u_i; it then constructs C = GRS_k(a,v') by scaling s = k-l coordinates and proves from Lemma 2.1 that the hull consists exactly of functions vanishing at those s points, giving dim Hull(C)=l. No parameter is fitted to a subset of the hull data, and no quantity is renamed as a prediction: the hull dimension is computed, not assumed. Theorem 2 uses the analogous Lemma 2.5 and the same degree argument with pi(x)=(x-b)^{m-k}; again the output is derived from, not identified with, the self-orthogonality condition. The examples import square-residue seed codes from the authors' earlier preprint [12]; this is a self-citation, but it is not circular because [12] supplies the independent existence of a self-orthogonal/self-dual seed satisfying stated field-theoretic conditions, and the theorem only needs such a seed as a hypothesis. The questionable direct derivation in Corollary 3.2(ii) -- where an even-length self-dual GRS seed is used as if it made the extended GRS code self-orthogonal -- is a proof gap, not a circular reduction; the hull-dimension mechanism itself does not presuppose the conclusion.

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

The theorems are conditional on a self-orthogonal seed code; the axioms are the background theory of GRS codes and the cited lemmas. The examples additionally import seed codes from [12], which is self-citation but not the target result.

assumptions (5)

standard math GRS_k(a,v) and its extended version are MDS and their duals are GRS codes (MacWilliams and Sloane, Chapter 11).
Used throughout Section 2 as the starting point for hull calculations.
domain assumption Lemma 2.1 and 2.2 from Chen and Liu [4] characterize the hull of GRS and extended GRS codes via polynomial identities.
Quoted and used as the computational engine in Theorems 1 and 2 to determine when a codeword lies in the hull.
standard math Lemma 2.3 from Li, Xing, and Wang [21]: the power sums Σ a_i^m u_i vanish for m ≤ n-2 and equal 1 for m = n-1.
Used in Lemma 2.5 to force the top coefficient of λ to be -1.
domain assumption For the examples, the existence of self-dual or almost self-dual GRS codes with the required square-residue properties is taken from the authors' preprint [12].
This is the seed input for Theorems 1 and 2 in the concrete families; it is not re-derived in this paper.
standard math The field size q satisfies q > 3, so there exists an element α with α^2 ≠ 1.
Needed in the proof to separate the scaled and unscaled coordinates; for q > 3 the group F_q^* has more than two elements.

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Pith. "Pith review of On Euclidean Hulls of MDS Codes." pith.science (2026). https://pith.science/paper/T7RNWVMM

@misc{pith2026190801173,
  author       = {Pith},
  title        = {Pith review of: On Euclidean Hulls of MDS Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7RNWVMM}},
  note         = {Machine review of arXiv:1908.01173}
}

read the original abstract

In this paper, we propose a mechanism on the constructions of MDS codes with arbitrary dimensions of Euclidean hulls. Precisely, we construct (extended) generalized Reed-Solomon(GRS) codes with assigned dimensions of Euclidean hulls from self-orthogonal GRS codes. It turns out that our constructions are more general than previous works on Euclidean hulls of (extended) GRS codes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [12]

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Fang, X., Lebed, K., Liu, H., Luo, J.: New MDS self-dual codes ove r ﬁnite ﬁelds of odd characteristic. arXiv:1811.02802v9 [cs.IT] Sep. 2019

work page Pith review arXiv 2019

[2] [1]

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Assmus, E.F., Key, J.D.: Designs and Their Codes. Cambridge Univer sity Press, Cambridge Tracts in Mathematics 103, Cambridge (1992)

work page 1992

[3] [2]

IEEE Trans

Blaum, M., Roth, R.M.: On lowest density MDS codes. IEEE Trans. In f. Theory, 45(1), 46-59 (1999)

work page 1999

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Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors wit h entanglement. Science, 314 436-439 (2006)

work page 2006

[5] [4]

IEEE Trans

Chen, B., Liu, H.: New constructions of MDS codes with complement ary duals. IEEE Trans. Inf. Theory, 64(8), 5776-5782 (2018)

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[8] [7]

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Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over Fq which are equivalent to LCD codes for q > 3. IEEE Trans. Inf. Theory, 64(4), 3010-3017 (2018)

work page 2018