{"id":"9bb890b0-956a-4641-b7d2-b277885dfec9","arxiv_id":"2506.00994","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs new families of self-orthogonal and self-dual algebraic geometric codes by generalizing residue criteria, yielding several quantum codes.","lead":"The paper proposes a residue-based criterion for when algebraic geometric codes are self-orthogonal, and uses it to build new families of self-dual and almost self-dual codes, plus quantum codes. Why read it: self-orthogonal codes with good distance parameters feed directly into quantum error correction, so the construction method matters for that field.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's advertised quantum parameters do not follow: the stated code dimension forces quantum dimension q^3−2q^2+q, not q^3−2q^2+q+4, and the claimed distance is a bound for C, not for C^⊥.","rationale":"I read the paper in good faith and checked the central residue criterion. Lemma 2 is a plausible generalization of Stichtenoth's criterion, and the divisor computations in Theorem 3 appear to be consistent with the Hermitian curve setup. The reader's stated weakest assumption, however, does not clearly land: in Theorem 4, using only floor((q−1)/m) cosets remains valid even when more cosets of ⟨α⟩ meet F_q^*, since the construction only needs λ ≤ floor((q−1)/m) cosets; the reader's q=4, m=3 example is also not a genuine counterexample because 3 divides q−1=3. I therefore cannot endorse that specific objection. The quantum-parameter inconsistency in Theorem 5 is a stronger and valid concern: the dimension forced by Riemann–Roch and the required dual distance are both incompatible with the stated parameters. This directly affects the advertised quantum-code applications, which are a central claim of the paper. Since the manuscript contains this and several other concrete parameter errors, the REJECT verdict stands, but my route to it differs from the reader's weakest-assumption diagnosis.","tokens_in":16701,"tokens_out":33806,"duration_ms":330655,"concrete_test":"Recompute Theorem 5 using the proof's r = q^2 − q − 1: k = l(rQ_∞) = q(q−1)/2, so the Hermitian construction gives quantum dimension q^3 − 2q^2 + q, and the only theorem-level lower bound on d(C^⊥_H) is deg(G') − (2g − 2) = 1. If the recomputation confirms this, the claimed [[q^3 − q^2, q^3 − 2q^2 + q + 4, ≥ q^3 − 2q^2 + q + 1]]_q parameters do not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing problem is in Section 4.2.2. Theorem 5 constructs C_L(D,G') with G' equivalent to rQ_∞ for r = q^2 − q − 1 = 2g − 1. Riemann–Roch then gives dim = l(rQ_∞) = r + 1 − g = q(q−1)/2 = g, so the Hermitian construction yields a quantum code of dimension n − 2k = (q^3 − q^2) − q(q−1) = q^3 − 2q^2 + q, not the stated q^3 − 2q^2 + q + 4. The stated code distance lower bound also disagrees with the proof's own n − r: for r = q^2 − q − 1, n − r = q^3 − 2q^2 + q + 1, while Theorem 5 says q^3 − 2q^2 + q − 1. Moreover, the quantum distance in the theorem is taken to be the lower bound for C_L(D,G') itself; the Hermitian construction requires the minimum distance of C^⊥_H, and Theorem 1 gives only d^⊥ ≥ deg(G') − (2g − 2) = 1 for this G'. No argument in the paper closes that gap. Because the abstract advertises quantum codes with notably good parameters, these arithmetic and distance errors are load-bearing: the stated Theorem 5 quantum parameters are not certified. Example 1 in Theorem 3 has the same kind of dimension slip (r = 181 gives k = 143, not 104), indicating the issue is systemic rather than an isolated typo.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a residue-based criterion (Lemmas 2 and 4) for constructing Euclidean and Hermitian self-orthogonal algebraic-geometric codes, and applies it to Artin-Schreier curves y^q+y=x^m and to Hermitian curves, with the aim of producing self-dual, almost self-dual, and quantum codes. The main advertised results are several families of self-orthogonal AG codes and quantum codes with good parameters, summarized in Tables 1-3.","tokens_in":16895,"tokens_out":1967,"duration_ms":18935,"significance":"The residue-criterion approach is a natural and potentially useful extension of Stichtenoth's self-dual Goppa code criterion, and the paper usefully collects several curves where the residue condition can be checked explicitly. The construction pipeline—checking a differential's polar divisor and residue cosets, then applying Lemma 2/4—is conceptually clean and, where the arithmetic is correct, produces valid self-orthogonal codes. However, the paper's central quantitative claims contain several arithmetic errors that affect the advertised code parameters, especially in Theorems 4, 5 and in Example 1. Because these parameters are load-bearing for the claimed quantum-code improvements, the paper in its current form does not certify its main results.","major_comments":[{"comment":"The proof of Lemma 4 contains an apparent typo that obscures the argument: the line '2G' − D = 2G − D − (q + 1)(u)' should presumably be '2G' − D = 2G − D − 2(u)' to match the subsequent use of u^{-2}η. If this is only a typo, the proof is otherwise consistent, but it should be corrected.","section":"Lemma 4"},{"comment":"The condition stated for the existence of the point set S is incorrect. The proof asserts that when m | (q^2−1) there are floor((q−1)/m) cosets of ⟨α⟩ with representatives in F_q^*. That count holds only when m | (q−1); when m divides q+1 but not q−1 (e.g., q=4, m=3), the number of such cosets is different and the later length formula q(λm+1) and the degree computations in (η) ≥ (q+1)Q_∞−D break down. The theorem should either impose m | (q−1) explicitly or provide a correct coset count for the m | (q+1) case.","section":"Theorem 4"},{"comment":"The quantum parameters in Theorem 5 are inconsistent with the stated construction. For r = q^2−q−1 = 2g−1, Riemann-Roch gives dim C_L(D,G') = r+1−g = q(q−1)/2 = g, so the Hermitian construction yields a quantum code of dimension n−2k = (q^3−q^2) − q(q−1) = q^3−2q^2+q, not q^3−2q^2+q+4. The claimed quantum distance also does not follow from the Hermitian construction: the construction requires a lower bound on the minimum distance of the Hermitian dual, and Theorem 1 gives d^⊥ ≥ deg(G')−(2g−2) = 1 for this choice of G'. The stated distance q^3−2q^2+q+1 is not justified.","section":"Theorem 5"},{"comment":"The numerical example attached to Theorem 3 contains a dimension error that suggests the issue is not isolated to Theorem 5. For q=27, m=7, r=181, the genus is g=(m−1)(q−1)/2 = 78, so the stated formula k0 = r−g+1 gives k0 = 104, which matches the printed value; however, a direct Riemann-Roch computation for deg(G') = r = 181 gives l(rQ_∞) = r+1−g = 181+1−78 = 104, so the claimed value 104 is correct only if r is indeed 181 and the genus is 78. The reported quantum dimension 4733 equals n−2k = 4941−208, which is consistent with k=104; the apparent discrepancy flagged by the stress-test does not arise if the printed k=104 is used. This example therefore does not by itself demonstrate a systemic dimension slip, but it is worth verifying the stated r = 181 satisfies the theorem's range mq−m−q ≤ r ≤ m(q−1)−1, i.e. 7·27−7−27 = 155 ≤ 181 ≤ 7·26−1 = 181, so the example is internally consistent. The concern about Theorem 5 remains valid regardless.","section":"Example 1"},{"comment":"Even if the dimension error in Theorem 5 were corrected, the claimed quantum distance lacks support. The Hermitian construction gives a quantum code with distance at least the minimum distance of C^{⊥_H}, not of C_L(D,G'). The proof only bounds the minimum distance of C_L(D,G') by n−r, and no separate lower bound for the dual is derived. The assertion that the quantum code has distance ≥ q^3−2q^2+q+1 is therefore unproved.","section":"Section 4.2.2 and quantum distance"}],"minor_comments":[{"comment":"The paper has numerous typographical and grammatical errors, including 'Hermtian' in the abstract, 'introduces' instead of 'introduce', inconsistent use of 'Euclidian' vs 'Euclidean', and repeated 'Throrem' in the tables. These should be corrected in a revision.","section":"Throughout"},{"comment":"The distinction between Lemma 2 and Lemma 4 is not clearly explained: Lemma 2 is stated for Hermitian self-orthogonality over F_{q^2}, while Lemma 4 is the Euclidean analogue. The proofs are similar but the notation 'f^q' in Lemma 4 appears without specifying the intended field automorphism; this should be clarified.","section":"Section 3"},{"comment":"The lower bound on the distance in Theorem 8(1) is stated as d0 ≥ n−r, but the table in Table 3 lists '≥ q(2t+1)p^{sl}−r' for the same theorem, which is inconsistent with the text's n = mq^2−mq+q. The table entries should be harmonized with the theorem statements.","section":"Theorem 8"},{"comment":"In the proof of Theorem 9(2), the equation '2r = q(k+1)+q(q−1)−2' uses the undefined symbol k; it should presumably be the parameter s from the theorem statement. This makes the self-dual condition unclear.","section":"Theorem 9"},{"comment":"The notation in Theorem 10 is confusing: the divisor D is defined using the set F_{q^2}\\U_k, but the parameter k is also used as the code dimension elsewhere. The conflict should be resolved by renaming one of the parameters.","section":"Theorem 10"},{"comment":"The example states q=9, k=70, giving q^3−qk = 729−630 = 99, but the condition (q+1)|k is satisfied since 10|70. The parameters [99,49,≥32] are then computed from the formula, but the derivation of the distance lower bound in the example is not fully shown; this is a presentation issue.","section":"Example 6"}],"recommendation":"reject","confidential_remarks":"The paper's core criterion (Lemma 2/4) appears sound as a generalization of Stichtenoth's criterion, but the main applications contain arithmetic errors that invalidate several advertised parameters. The errors in Theorem 4 and Theorem 5 are not mere typos: they affect the existence of the constructed point set and the quantum code parameters, respectively. Given that the abstract and tables advertise these specific parameters as the main contribution, a major revision would be needed to re-derive correct parameters and re-verify each theorem. In my assessment, the current manuscript does not meet the standard for publication in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is worth taking seriously. Lemma 2 generalizes Stichtenoth's residue criterion to the Hermitian setting, and Lemma 4 does the same for squares/non-squares in the Euclidean case. That is a real contribution, not just a repackaging. The lifting construction in Theorem 6, from self-dual EGRS codes to self-dual AG codes, is also a nice idea, and the families on y^q + y = x^m for general m | q+1 genuinely extend Jin's m=3 work.\n\nBut the paper as written has too many load-bearing arithmetic errors. The most serious is Theorem 5. The Hermitian construction gives quantum dimension n-2k = q^3 - 2q^2 + q, not the stated q^3 - 2q^2 + q + 4. And the quantum distance is taken as a lower bound for C, but the construction requires a lower bound for C^⊥; for r = q^2 - q - 1, Theorem 1 gives only d^⊥ ≥ 1. So the advertised quantum parameters are not proved. Theorem 4 also has a condition gap: for m | q^2 - 1 but m ∤ q-1, the roots of unity lie outside F_q, and the residue computation that puts residues in F_q collapses. The condition should be m | q-1, or the proof needs a different argument. The examples also need fixing: Example 5 uses s=338, which does not divide q^2-1 = 624, and Example 6 says q is even while using q=9. Table 3 has wrong cross-references and a copied distance formula.\n\nOne small point: the stress-test's claim about Example 1 is off—r=181 with g=78 gives k = 181 - 78 + 1 = 104, which matches the paper. So that specific example is fine.\n\nThe residue criteria themselves are plausible and the divisor computations in Theorem 3 check out. The paper is for specialists in AG codes and quantum stabilizer codes. It deserves a serious referee—the central technique is worth developing—but it needs a careful revision: fix the conditions in Theorem 4, recompute the quantum parameters in Theorem 5, correct the examples, and re-verify the affected parameter claims. I'd send it to review, not desk-reject, but with a clear expectation of major revision.","headline":"The residue criterion is a genuine extension of Stichtenoth's method, but Theorem 5's quantum parameters don't follow and Theorem 4's condition is too weak; needs major revision.","tokens_in":744,"tokens_out":1024,"would_cite":false,"duration_ms":63298,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B27","11T71","14G50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a residue-based criterion under which an algebraic-geometric code admits an equivalent divisor making it Hermitian or Euclidean self-orthogonal, and uses that criterion to construct self-dual, almost self-dual, and…","keywords":["algebraic geometry codes","Hermitian self-orthogonal codes","Euclidean self-orthogonal codes","self-dual codes","quantum codes","maximal curves","residue criterion"],"falsifier":"Set $q=8$ and $m=3$. Since $\\mathbb{F}_8^*$ has seven elements and the subgroup $\\langle\\alpha\\rangle$ of order $3$ intersects it trivially, the cosets $t\\langle\\alpha\\rangle$ for $t\\in\\mathbb{F}_8^*$ are seven distinct cosets meeting $\\mathbb{F}_8^*$, not $\\lfloor(8-1)/3\\rfloor=2$. This direct count contradicts the counting premise of Theorem 4 under its stated hypotheses, so the length formula $q(\\lambda m+1)$ must be re-derived in this case.","tokens_in":16322,"feed_emoji":"🧮","tokens_out":17532,"duration_ms":150867,"temperature":0.7,"pith_summary":"The paper tries to show that self-orthogonality of an algebraic-geometric (AG) code can be read off from the residues of one differential, and that this test is powerful enough to construct new families of self-dual, almost self-dual, and quantum codes. The central result says: if a differential $\\eta$ on the curve has divisor $(q+1)G-D$ and its residues at the evaluation places all lie in the same coset of $\\mathbb{F}_q^*$, then the code $C_L(D,G)$ can be replaced by an equivalent code $C_L(D,G')$ that is Hermitian self-orthogonal; replacing $(q+1)G-D$ by $2G-D$ and requiring all residues square or non-square gives the Euclidean analogue. A sympathetic reader should care because AG codes can outperform random codes at the same length, and self-orthogonal AG codes can be converted into quantum codes by the standard Hermitian construction. The paper applies the test to maximal curves of the form $y^q+y=x^m$ and to Hermitian curves, producing codes with parameters close to the optimal distance bound and, for even $q$, Euclidean self-dual codes of several new lengths.","feed_headline":"Residue rule yields self-orthogonal and quantum codes","feed_subtitle":"Matching residues put algebraic-geometry codes inside their duals, yielding self-dual and quantum codes","key_machinery":"The machinery is a residue differential plus an equivalent-divisor adjustment. Given $D=P_1+\\cdots+P_n$ and a divisor $G$ disjoint from $D$, one chooses $\\eta$ whose residues at $P_i$ lie in one coset of $\\mathbb{F}_q^*$ and whose divisor is large enough. A function $u$ obtained from the Chinese Remainder Theorem then adjusts $G$ to $G'=G-(u)$, making all residues equal to $1$; Theorem 2, the duality theorem $C_\\Omega(D,G)=C_L(D,D-G+(\\eta))$, converts this into the inclusion that defines self-orthogonality. The specific constructions use $\\eta=dx/(x(x^{m(q-1)}-1))$ on the curve $y^q+y=x^m$, where the product identity $\\prod_{j\\ne i}(\\alpha^i-\\alpha^j)=m\\alpha^{-i}$ forces the residues to be constant, and analogous differentials $dx/\\prod(x-d)$ on Hermitian curves. The Riemann-Roch formula then gives the dimensions of the resulting codes.","core_discovery":"On the paper's own terms, the central discovery is a divisorial residue criterion: Lemma 2 states that for an AG code $C_L(D,G)$ over $\\mathbb{F}_{q^2}$, if there is a differential $\\eta$ with $(\\eta)\\ge (q+1)G-D$ and with residues $\\mathrm{res}_{P_i}(\\eta)$ all contained in one coset of $\\mathbb{F}_q^*$, then some divisor $G'$ linearly equivalent to $G$ makes $C_L(D,G')$ Hermitian self-orthogonal. Lemma 4 is the Euclidean counterpart: $(\\eta)\\ge 2G-D$ together with all residues square or all non-square yields Euclidean self-orthogonality, and equality in the divisor condition yields self-duality. The proof works by using the Chinese Remainder Theorem to multiply $\\eta$ by a function that equalizes all residues to $1$, so that the dual description $C_\\Omega(D,G)=C_L(D,D-G+(\\eta))$ puts the code inside its dual. The rest of the paper applies this criterion to specific maximal curves, and separately lifts self-dual generalized Reed-Solomon codes into (almost) self-dual AG codes.","pith_inferences":["The same residue criterion should apply to any Artin-Schreier or Kummer extension in which $dx/(x^a-b)$ has constant residues at a chosen set of rational places; testing the criterion on other maximal curves with known rational point distributions is a direct next step.","If the coset count in Theorem 4 is repaired, the same Hermitian-curve construction would allow more multiplicative cosets, potentially producing code lengths larger than $q(\\lambda m+1)$; for $q=8,m=3$, seven cosets meet $\\mathbb{F}_8^*$, not two.","The lifting construction suggests a general correspondence: every self-dual MDS code of length $n$ over $\\mathbb{F}_{q^2}$ may be lifted to a self-dual AG code of length $\\lambda n$ whenever a curve has $\\lambda$ completely split places above each evaluation point; cataloging which curves admit such lifts is a natural testable program.","Because the equality case of the divisorial condition gives self-duality, a similar search for differentials with $(\\eta)=2G-D$ on curves with many rational places should yield Euclidean self-dual AG codes for odd $q$ too, where parity would then not be an obstruction."],"forward_implications":["For the curve $y^q+y=x^m$ with $m\\mid q+1$ and $r$ in the stated range, Theorem 3 produces Hermitian self-orthogonal $[mq^2-mq+q,\\,r-\\frac12(m-1)(q-1)+1,\\,\\ge mq^2-mq+q-r]_{q^2}$ codes and quantum $[[mq^2-mq+q,\\,mq^2-m-2r-1,\\,\\ge r-mq+m+q+1]]_q$ codes.","For the Hermitian curve $y^q+y=x^{q+1}$, Theorem 5 gives a Hermitian self-orthogonal code with parameters $[q^3-q^2,\\,\\frac{q(q-1)}2,\\,\\ge q^3-2q^2+q-1]_{q^2}$ and quantum codes with distance at least $q^3-2q^2+q+1$.","The Euclidean version yields Euclidean self-dual codes for even $q$ in several families, with lengths $mq^2-mq+q$, $q(s+1)$, $q^3-qk$, and $p^kq$; for odd lengths it yields almost self-dual codes of dimension $(n-1)/2$.","Theorem 6 lifts any self-dual extended generalized Reed-Solomon (EGRS) code of length $2t+2$ to an (almost) self-dual AG code of length $q(2t+1)p^{sl}$ on any curve with enough completely split places, so known MDS self-dual code families translate into longer AG codes.","These constructions generalize the earlier quantum codes from maximal curves with $q$ an odd power of two and $m=3$ to all $m\\mid q+1$, and the resulting quantum codes have larger distances than the comparison families in the tables."],"supporting_citations":[{"why":"Supplies the foundational duality theorem $C_\\Omega(D,G)=C_L(D,D-G+(\\eta))$ and the Riemann-Roch parameter formulas used throughout the residue criterion.","marker":"[26]"},{"why":"Provides the product identity and the EGRS self-duality lemma that force constant residues and drive the generic GRS-to-AG lifting construction.","marker":"[33]"},{"why":"States the earlier Euclidean square-residue criterion that Lemmas 3 and 4 generalize to the Hermitian and non-square settings.","marker":"[27]"},{"why":"Gives the generalized Reed-Solomon self-duality criterion used as the seed for the generic AG construction.","marker":"[18]"},{"why":"Establishes the quantum stabilizer codes from maximal curves that Theorem 3 generalizes and that the tables compare against.","marker":"[15]"},{"why":"Supplies the Hermitian construction that turns Hermitian self-orthogonal codes into quantum codes, the paper's main application route.","marker":"[1]"}],"fun_headline_variants":["Residue criterion for self-orthogonal AG codes","Residue matching makes AG codes self-dual","Generic construction of self-orthogonal AG codes","Residue equalization yields self-orthogonal codes","Quantum codes from self-orthogonal AG construction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction in Theorem 4 assumes that, for a primitive $m$-th root of unity $\\alpha\\in\\mathbb{F}_{q^2}$ with $m\\mid(q^2-1)$, there are exactly $\\lfloor(q-1)/m\\rfloor$ cosets of $\\langle\\alpha\\rangle$ with representatives in $\\mathbb{F}_q^*$; this counting is guaranteed only when $m\\mid q-1$, while the theorem only assumes $m\\le q-1$ and $m\\mid q^2-1$.","fun_headline_variants_meta":{"raw":{"variants":["Residue criterion for self-orthogonal AG codes","Residue matching makes AG codes self-dual","Generic construction of self-orthogonal AG codes","Residue equalization yields self-orthogonal codes","Quantum codes from self-orthogonal AG construction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1571,"prompt_tokens":930,"completion_tokens":641,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":565}},"tokens_in":546,"tokens_out":641,"duration_ms":5788,"temperature":1.0,"reasoning_tokens":565,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:56:45.884595+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $q=8$ and $m=3$. Since $\\mathbb{F}_8^*$ has seven elements and the subgroup $\\langle\\alpha\\rangle$ of order $3$ intersects it trivially, the cosets $t\\langle\\alpha\\rangle$ for $t\\in\\mathbb{F}_8^*$ are seven distinct cosets meeting $\\mathbb{F}_8^*$, not $\\lfloor(8-1)/3\\rfloor=2$. This direct count contradicts the counting premise of Theorem 4 under its stated hypotheses, so the length formula $q(\\lambda m+1)$ must be re-derived in this case.","supporting_citations":[{"cited_title":"Stichtenoth, Algebraic function fields and codes(Graduate Texts in Mathematics), vol","cited_arxiv_id":null,"evidence_quote":"Supplies the foundational duality theorem $C_\\Omega(D,G)=C_L(D,D-G+(\\eta))$ and the Riemann-Roch parameter formulas used throughout the residue criterion."},{"cited_title":"A note on the constructions of MDS self-dual codes,","cited_arxiv_id":null,"evidence_quote":"Provides the product identity and the EGRS self-duality lemma that force constant residues and drive the generic GRS-to-AG lifting construction."},{"cited_title":"Self-dual Goppa codes,","cited_arxiv_id":null,"evidence_quote":"States the earlier Euclidean square-residue criterion that Lemmas 3 and 4 generalize to the Hermitian and non-square settings."},{"cited_title":"New MDS self-dual codes from generalized Reed-Solomon codes,","cited_arxiv_id":null,"evidence_quote":"Gives the generalized Reed-Solomon self-duality criterion used as the seed for the generic AG construction."},{"cited_title":"Quantum stabilizer codes from maximal curves,","cited_arxiv_id":null,"evidence_quote":"Establishes the quantum stabilizer codes from maximal curves that Theorem 3 generalizes and that the tables compare against."},{"cited_title":"Nonbinary quantum stabilizer codes,","cited_arxiv_id":null,"evidence_quote":"Supplies the Hermitian construction that turns Hermitian self-orthogonal codes into quantum codes, the paper's main application route."}],"review_version":1}