{"id":"95492739-9a57-487c-a96e-b11c4eded2b5","arxiv_id":"1908.01173","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Any self-orthogonal (extended) GRS code of dimension m yields MDS codes of every smaller dimension k with any Euclidean hull dimension from 0 to k.","lead":"This paper gives a general method to build MDS error-correcting codes with any desired hull size, starting from any self-orthogonal Reed-Solomon code. The result expands the toolkit for designing quantum error-correcting codes that use entanglement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main Theorems 1 and 2 are correct, but Corollary 3.2(ii) is not a direct consequence of Theorem 2: the even-length self-dual seed does not give a self-orthogonal extended GRS code, so the proof needs correction.","rationale":"The reader accepted the paper with high confidence and flagged only the reliance on [12] for examples. My independent check confirms Theorems 1 and 2: the construction scales coordinates by alpha with alpha^2 != 1, and the degree bounds force the polynomial identities needed to compute the hull dimension. The problem I found is internal to Corollary 3.2(ii): the proof claims a direct application of Theorem 2 to a self-dual GRS_{n/2}(a,v), but Lemma 2.5 shows the corresponding extended GRS code is not self-orthogonal unless v_i^2 = (lambda_0 - a_i) u_i. This is not a fatal flaw in the main mechanism, because a corrected argument with the same alpha-scaling and a polynomial pi(x) proves the corollary. It is, however, a genuine unsupported step in a stated result and in Example 4.1(ii). I therefore recommend conditional acceptance pending a corrected proof of Corollary 3.2(ii).","tokens_in":13905,"tokens_out":31044,"duration_ms":282706,"concrete_test":"Set n = 2m and assume v_i^2 = lambda_0 u_i. Use Lemma 2.2 to compute Hull(C) for C = GRS_k(a,v',infinity) with v'_i = alpha v_i pi(a_i) for i <= s and v_i pi(a_i) for i > s, where pi(x) = (x-b)^{m-k}. Check that lambda_0 pi^2 f has degree at most n-k-1 while g has degree at most n-k, so the leading-coordinate equality forces f_{k-1} = 0; then count the dimension of polynomials f vanishing at the first s evaluation points. If the count is k-s-1, the correct choice of s in Corollary 3.2(ii) is s = k-l-1, not the s = k-l used in Theorem 2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The mechanism in Theorems 1 and 2 checks out: the degree counts in Lemmas 2.4 and 2.5 and the polynomial identity arguments are valid. The load-bearing gap is in the derivation of Corollary 3.2(ii). For n even, self-duality of GRS_{n/2}(a,v) means v_i^2 = lambda_0 u_i with lambda_0 constant, by Corollary 2.1. But the proof says the result follows from Theorem 2 by choosing this self-dual code. To invoke Theorem 2, one would need the extended code GRS_{n/2}(a,v,infinity) to be self-orthogonal, and Lemma 2.5 requires v_i^2 = (lambda_0 - a_i) u_i, which is not constant. Hence the stated direct derivation is invalid; the extended seed is not self-orthogonal in general. The same gap propagates to Example 4.1(ii), which relies on Corollary 3.2(ii). The corollary is salvageable: with pi(x) = (x-b)^{m-k} and s = k-l-1 (with s = k for l = 0), the hull computation via Lemma 2.2 gives dim Hull(C) = l, so the statement appears true but the paper's proof is incomplete.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14104,"tokens_out":7471,"duration_ms":62509,"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":[{"comment":"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":"Section 3, Corollary 3.2(ii) and Section 4, Example 4.1(ii)"},{"comment":"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].","section":"Section 4, Examples 4.1–4.3"}],"minor_comments":[{"comment":"The abstract contains typographical artifacts: 'MD S codes' and 'g eneralized' should be 'MDS codes' and 'generalized'.","section":"Abstract"},{"comment":"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":"Remark 3.3(iii)"},{"comment":"The phrase 'linear independent' should be 'linearly independent'.","section":"Section 2, proof of Lemma 2.4"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are sound, but the proof of Corollary 3.2(ii) is incorrect as written. Since the corollary is one of the advertised results, the paper requires a major revision rather than acceptance. The dependence on the authors' unpublished preprint [12] for the examples also merits editorial attention."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":14725,"tokens_out":1900,"would_cite":true,"duration_ms":19814,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B27","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"From any self-orthogonal generalized Reed-Solomon code, this paper builds maximum distance separable codes with every prescribed Euclidean hull dimension.","keywords":["MDS codes","Euclidean hull","generalized Reed-Solomon codes","self-orthogonal codes","self-dual codes","hull dimension","entanglement-assisted quantum error-correcting codes"],"falsifier":"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 $i$.","tokens_in":13637,"feed_emoji":"⚛️","tokens_out":25843,"duration_ms":186154,"temperature":0.7,"pith_summary":"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.","feed_headline":"Self-orthogonal GRS codes yield every MDS hull dimension","feed_subtitle":"The hull size controls the entanglement cost of the resulting quantum codes.","key_machinery":"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 $k-s$.","core_discovery":"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 $k-s = l$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the characterizations (Lemmas 2.1 and 2.2) of when a codeword lies in the hull of a GRS code, which the proof uses to count hull dimensions.","marker":"[4]"},{"why":"Provides the square-residue results guaranteeing the self-dual and almost self-dual GRS codes used in Examples 4.1-4.3.","marker":"[12]"},{"why":"Supplies the power-sum identity (Lemma 2.3) used to determine the polynomial lambda in the self-orthogonality characterizations.","marker":"[21]"},{"why":"Gives the earlier sufficient condition for GRS self-duality that Lemma 2.4 sharpens into a necessary-and-sufficient criterion.","marker":"[19]"},{"why":"Gives the earlier construction of MDS codes with hulls of arbitrary dimensions that Corollary 3.1 generalizes.","marker":"[23]"},{"why":"Provides earlier hull constructions for (extended) GRS codes that Corollaries 3.1 and 3.2 recover as special cases.","marker":"[11]"}],"fun_headline_variants":["MDS codes with any Euclidean hull dimension","Arbitrary hull dimensions for MDS codes","Self-orthogonal GRS codes yield all hull dimensions","Every hull dimension for MDS codes","Explicit construction: MDS codes with any hull"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["MDS codes with any Euclidean hull dimension","Arbitrary hull dimensions for MDS codes","Self-orthogonal GRS codes yield all hull dimensions","Every hull dimension for MDS codes","Explicit construction: MDS codes with any hull"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000673,"raw_usage":{"total_tokens":3043,"prompt_tokens":904,"completion_tokens":2139,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":2078}},"tokens_in":520,"tokens_out":2139,"duration_ms":16821,"temperature":1.0,"reasoning_tokens":2078,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:21:53.390905+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $i$.","supporting_citations":[{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the characterizations (Lemmas 2.1 and 2.2) of when a codeword lies in the hull of a GRS code, which the proof uses to count hull dimensions."},{"cited_title":"New MDS Self-dual Codes over Finite Fields of Odd Characteristic","cited_arxiv_id":"1811.02802","evidence_quote":"Provides the square-residue results guaranteeing the self-dual and almost self-dual GRS codes used in Examples 4.1-4.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the power-sum identity (Lemma 2.3) used to determine the polynomial lambda in the self-orthogonality characterizations."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Gives the earlier sufficient condition for GRS self-duality that Lemma 2.4 sharpens into a necessary-and-sufficient criterion."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Gives the earlier construction of MDS codes with hulls of arbitrary dimensions that Corollary 3.1 generalizes."},{"cited_title":"Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs","cited_arxiv_id":"1812.09019","evidence_quote":"Provides earlier hull constructions for (extended) GRS codes that Corollaries 3.1 and 3.2 recover as special cases."}],"review_version":1}