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REVIEW 5 major objections 6 minor 36 references

A Generic Construction on Self-orthogonal Algebraic Geometric Codes and Its Applications

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.00994 v2 pith:JW7WPKO3 submitted 2025-06-01 cs.IT math.IT

Puyin Wang , Jinquan Luo This is my paper
classification cs.ITmath.IT MSC 94B2711T7114G50
keywords algebraicgeometrycodesHermitianself-orthogonalEuclideanself-dualquantummaximalcurvesresiduecriterion
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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 $m\mid q^2-1$.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

5 major / 6 minor

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.

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 (5)
  1. [Lemma 4] 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.
  2. [Theorem 4] 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.
  3. [Theorem 5] 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.
  4. [Example 1] 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.
  5. [Section 4.2.2 and quantum distance] 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.
minor comments (6)
  1. [Throughout] 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.
  2. [Section 3] 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.
  3. [Theorem 8] 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.
  4. [Theorem 9] 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.
  5. [Theorem 10] 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.
  6. [Example 6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the self-orthogonality criteria are proved from Stichtenoth's duality theorem, and all constructions verify those criteria directly; the few self-citations are background and not load-bearing.

full rationale

The paper's central derivation chain is self-contained. Lemma 2 and Lemma 4 are proved from Theorem 2 (Stichtenoth's duality C_Ω(D,G)=C_L(D,D−G+(η))) by constructing a function u via the Chinese Remainder Theorem and setting G′=G−(u); the self-orthogonality conclusion follows from the residue theorem, not from assuming the target result. The main constructions (Theorems 3, 5, 8–11) proceed by explicitly computing the divisor (η) and the residues res_{P_i}(η) for a chosen differential on a specific curve, then verifying the hypotheses of Lemma 2 or Lemma 4. For Theorem 6 and Theorem 7, the paper takes the existence of a self-dual GRS/EGRS code as an input (from Lemmas 5–6, due to [18] and [33]) and uses only the known residue criteria to lift the construction to AG codes; this is a legitimate use of external results, not an import of the paper's own conclusion. The self-citations [9] and [32] are mentioned only as background on MDS self-dual code construction and as motivation for generalizing to AG codes; the actual proof does not rely on these papers as black boxes. No fitting of parameters is disguised as prediction, and no uniqueness theorem is imported from the authors' prior work. The manuscript contains apparent arithmetic slips (e.g., the dimension in Example 1 and the quantum dimension in Theorem 5), but these are correctness errors, not instances of circular reasoning: the claimed parameters do not reduce to the inputs by definition. Therefore the circularity score is 0.

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

The constructions rely on standard theorems of algebraic function fields (Riemann-Roch, Hasse-Weil, AG code duality) and on prior criteria for GRS and EGRS self-duality and the Hermitian quantum construction. No new entities or fitted constants are introduced; the integer parameters m, r, s, k are existence parameters in the theorems, not fitted values.

assumptions (5)
  • standard math Riemann-Roch theorem
    Used to compute dimensions of L(rQ∞) in Theorems 3, 5, 8-11.
  • standard math Hasse-Weil bound
    Used to show the Artin-Schreier curve y^q + y = x^m with m | q+1 is maximal over F_{q^2}.
  • standard math AG code duality C_Ω(D,G) = C_L(D, D-G+(η)) from Stichtenoth [26]
    Basis for Lemmas 2-4.
  • domain assumption GRS and EGRS self-duality criteria (Lemma 5 from [18], Lemma 6 from [33])
    Input for the lifting construction in Theorems 6-7.
  • domain assumption Hermitian construction for quantum codes from [1]
    Used to convert Hermitian self-orthogonal codes into quantum codes.

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Pith. "Pith review of A Generic Construction on Self-orthogonal Algebraic Geometric Codes and Its Applications." pith.science (2026). https://pith.science/paper/JW7WPKO3

@misc{pith2026250600994,
  author       = {Pith},
  title        = {Pith review of: A Generic Construction on Self-orthogonal Algebraic Geometric Codes and Its Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JW7WPKO3}},
  note         = {Machine review of arXiv:2506.00994}
}
read the original abstract

In the realm of algebraic geometric (AG) codes, characterizing dual codes has long been a challenging task. In this paper we introduces a generalized criterion to characterize self-orthogonality of AG codes based on residues, drawing upon the rich algebraic structures of finite fields and the geometric properties of algebraic curves. We also present a generic construction of self-orthogonal AG codes from self-dual MDS codes. Using these approaches, we construct several families of self-dual and almost self-dual AG codes. These codes combine two merits: good performance as AG code whose parameters are close to the Singleton bound together with Euclidean (or Hermtian) self-dual/self-orthogonal property. Furthermore, some AG codes with Hermitian self-orthogonality can be applied to construct quantum codes with notably good parameters.

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