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Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Non-special divisors on hyperelliptic curves reduce to a degree check on one polynomial.

desk verdict The Mumford-degree test for non-special divisors is a real and useful idea, but the abstract overclaims the MDS LCD examples: only q=5 matches the advertised parameters, and the q=4 example fails the paper's own 2-torsion hypothesis. read the letter →

arxiv 2607.16945 v1 pith:GSRYWRGD submitted 2026-07-18 math.AG cs.ITmath.IT

Adler Marques , Yuri da Silva , Saeed Tafazolian This is my paper
classification math.AGcs.ITmath.IT MSC 11G2014H4014G5094B27
keywords hyperellipticcurvesMumfordrepresentationnon-specialdivisorslinearcomplementarypairsLCDcodesMDSalgebraicgeometrymaximal
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 tries to establish that on a hyperelliptic curve of genus g, deciding whether a divisor of degree g−1 or g is non-special—a classical geometric question equivalent to asking whether the divisor class avoids the theta divisor (the subvariety of special classes) in the Jacobian—collapses to a one-line test in the Mumford representation (a pair of polynomials encoding each divisor class): the reduced representative's u-polynomial must have degree exactly g (for degree g−1) or at least g−1 (for degree g). Using this test, the paper constructs linear complementary pairs (LCPs) of algebraic-geometry codes by Chinese-remainder arithmetic on Mumford pairs, gives a criterion for those codes to be maximum distance separable (MDS), and, under a 2-torsion condition expressed as u_A | (2v_A + h), derives an explicit multiplier vector that turns an LCP into an LCD code. The framework is applied to the maximal hyperelliptic curve y² = x^q + x over F_{q²}, with computationally verified examples for q = 4, 5, 7 and a conjecture that [2q, q, q+1]_{q²} MDS LCD codes exist for every q ≥ 4. A sympathetic reader would care because explicit, checkable non-special divisors were a known bottleneck for moving LCP/LCD constructions beyond genus zero, and MDS parameters are the best possible for a given length and dimension.

What carries the argument

The central object is the reduced Mumford representation of divisor classes on a hyperelliptic curve: every class in the Jacobian has a unique representative as an effective affine divisor of degree d ≤ g encoded by a monic polynomial u(x) = ∏(x − x_i)^{m_i} and a polynomial v with deg v < deg u and u | v² + hv − f. The load-bearing mechanism is that non-speciality of small-degree divisors is detected by deg u reaching its maximum value g (or g−1), replacing an intractable theta-avoidance decision with a univariate degree test. The constructions then combine this test with Chinese-remainder interpolation of the two Mumford pairs, the relation P + ι(P) ∼ 2P_∞, and the 2-torsion condition u_A

What would settle it

Compute directly, over F_{16}, the Euclidean dual of the code C_L(D,G) from Example 2 and compare it with m·C_L(D,H) using the multiplier formula of Theorem 4; the q=4 example does not satisfy u_A | 2v_A + h (h = 1 so 2v_A + h is constant), so a mismatch would show the explicit dual formula cannot be extended beyond the 2-torsion hypothesis.

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Extended reading notes

Core claim

The paper's central claim: for A of degree g−1, with (u_A, v_A) the reduced Mumford representative of [A − (g−1)P_∞], A is non-special iff deg u_A = g; for degree g, non-special iff deg u_A ≥ g−1. This turns theta-avoidance into a degree check. From two coprime Mumford pairs the paper constructs G = D_A + B − P_∞ and H = D_A + ι(B) − P_∞, whose gcd is the non-special A, so C_L(D,G) and C_L(D,H) form an LCP of length 2t and dimension t; the pair is MDS iff every class α±β−γ_I (I a t-subset of the evaluation support) has Mumford u-degree g. Under 2[D_A − gP_∞] = 0 (u_A | 2v_A + h), the dual equals m·C_L(D,H) with m_P = 1/(p'_D(x_P)(2y_P + h(x_P))u_A(x_P)u_B(x_P)), and square-root rescaling giv

Load-bearing premise

The LCD construction rests entirely on the 2-torsion condition u_A | 2v_A + h (equivalently 2[D_A − gP_∞] = 0 in the Jacobian), which forces every point of D_A to be ramified; when it fails—as in the paper's q=4 example, where h = 1 makes 2v_A + h constant—the explicit dual formula of Theorem 4 does not apply.

Editorial extensions

If this is right

  • The degree test gives an explicit, algorithmic route to non-special divisors of degree g−1 and g on every hyperelliptic curve, replacing Riemann-Roch dimension computations with a single polynomial-degree check.
  • For any evaluation divisor D of 2t distinct rational points forming t conjugate pairs and coprime to the Mumford data, the CRT construction yields an LCP of AG codes of length 2t, dimension t, and minimum distance at least t − g + 1.
  • The MDS property of both codes is equivalent to a finite list of Jacobian class-avoidance conditions, each checkable by Mumford u-degree, so the optimal-distance test becomes a routine computation.
  • Under the 2-torsion condition u_A | 2v_A + h, the dual of C_L(D,G) is exactly m·C_L(D,H) for the explicit multiplier m, and coordinate-wise square roots of m turn the LCP into an LCD code.
  • On the maximal curve y² = x^q + x over F_{q²}, the paper gives computational evidence for MDS LCP parameters [8,4,5]_{16}, MDS LCD parameters [10,5,6]_{25}, and LCD parameters [28,14]_{49}, and conjectures that [2q,q,q+1]_{q²} MDS LCD codes exist for all q ≥ 4.

Reading between the lines

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

  • The same Mumford-degree test suggests a deterministic polynomial-time routine for deciding non-speciality on hyperelliptic curves, since the underlying Mumford arithmetic runs in polynomial time in the genus and field size; the paper uses the test as a verification loop but does not state a complexity bound.
  • The paper's own examples support the abstract's [2q,q,q+1] MDS LCD claim at q=5 only: q=4 yields an MDS LCP (not an LCD code), and q=7 is run with k=2 giving [28,14]_{49}, so the stated conjecture currently rests on fewer verified parameter sets than the abstract suggests.
  • Because the 2-torsion condition forces D_A to be ramified, it is restrictive when h ≠ 0; on the y² = x^q + x family h = 0 makes the condition vacuous, so the multiplier construction may need a different mechanism (e.g., Hilbert-symbol-type square roots rather than ordinary field square roots) to extend to general hyperelliptic curves.
  • The MDS criterion is a finite check of binomial size—70 subsets for q=4 and 252 for q=5—which the paper verifies by computer; a natural extension is to turn the heuristic inequality 2q < (q+1)^{q−1} on the maximal curve into a counting proof that the forbidden Jacobian classes cannot exhaust the complement.
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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

2 major / 4 minor

Summary. The paper studies algebraic geometry codes on hyperelliptic curves of genus g≥2, with a focus on linear complementary pairs (LCP) and linear complementary dual (LCD) codes. The main theoretical contribution is a characterization of non-special divisors of degree g−1 and g in terms of the degree of the u-polynomial in their reduced Mumford representation (Proposition 3, Corollary 1, Theorem 1). Using this, the authors construct LCPs of codes via Chinese remainder arithmetic on Mumford pairs, give a polynomial-degree MDS criterion (Proposition 6, Theorem 3), and, under a 2-torsion condition u_A | 2v_A+h, derive an explicit multiplier that turns the LCP into an LCD code (Theorem 4). The framework is applied to maximal hyperelliptic curves over F_{q^2}, with Magma-verified examples for q=4,5,7 and a conjecture, with heuristic support, that MDS LCD codes of parameters [2q,q,q+1]_{q^2} exist for all q≥4.

Significance. The core theoretical results appear sound and useful: Proposition 3 and Theorem 1 give an explicit, checkable polynomial-degree criterion for non-speciality, replacing a hard geometric theta-avoidance problem with a univariate degree test. The CRT-based LCP construction and the explicit dual formula in Theorem 4 are elegant and constitute a genuine new tool for constructing complementary AG codes. The computational verification of the MDS conditions in the examples is a positive feature. However, the paper's headline claim of MDS LCD codes for q=4,5,7 is not supported by the examples as written: only the q=5 example realizes the advertised parameters. This is a substantive gap between the abstract and the demonstrations, and needs to be addressed before publication.

major comments (2)
  1. [Abstract; §1 and §4.2, Examples 2–4] The abstract and §1 claim explicit MDS LCD codes with parameters [2q,q,q+1]_{q^2} for q=4,5,7. This is not what the examples show. Example 2 (q=4) constructs only an LCP of MDS codes [8,4,5]_{16}; it is not shown to be LCD. Theorem 4’s hypothesis u_A | 2v_A+h cannot hold there: in char 2 with h=1, 2v_A+h=1, and u_A has degree 2; moreover #J(X)=5^4 is odd, so the required 2-torsion class does not exist. Example 4 (q=7) uses k=2 and produces an LCD [28,14]_{49}, not [14,7]_{49}, and its minimum distance is not computed. Only Example 3 (q=5) realizes the advertised MDS LCD parameters [10,5,6]_{25}. This overclaim is load-bearing: the paper’s headline is precisely these examples. Please either revise the abstract/introduction to the supported cases or supply the missing q=4 and q=7 k=1 constructions.
  2. [§4.2, Theorem 4 and Proposition 7] Theorem 4’s LCD construction is conditional on u_A | 2v_A+h, which forces every point of D_A to be ramified. This is a strong hypothesis and, as Example 2 shows, it is not satisfiable for the maximal hyperelliptic curve y^2+y=x^5 over F_16 (or any curve whose Jacobian has odd order). The paper should state this limitation explicitly when advertising the examples, since the q=4 example cannot be converted to an LCD code by the given method.
minor comments (4)
  1. [§4.1, Theorem 3] Theorem 3 states parameters [2t,t,t+1]_q, but the application in §4.2 uses alphabet F_{q^2}. Re-label the field in the theorem to avoid confusion.
  2. [§4.2, Theorem 4] The sentence 'which is always possible over a suitable extension or by careful choice of x_i' is imprecise; changing the extension changes the alphabet. The square condition is verified in Proposition 7; please clarify that the LCD code is over F_{q^2}.
  3. [§4.2, Example 4] Example 4 states that the chosen pairs 'satisfy the conditions of Proposition 7' but does not show the verification of the square condition m_a ∈ (F_{49}^*)^2 or list the multiplier µ. For reproducibility, either include the check or provide the Magma script.
  4. [§4, Remark 4] The counting heuristic only counts forbidden Jacobian classes for the MDS condition; it does not account for the square condition of Proposition 7, so it should not be read as supporting the LCD/MDS conjecture without additional argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Mumford-degree test, MDS criterion, and dual/LCD formulas are derived from Riemann–Roch and residue computations rather than assumed; self-citations are contextual only.

full rationale

The paper's central derivation chain is self-contained. Proposition 3 and Theorem 1 convert non-speciality into a degree condition on reduced Mumford representatives by a direct Riemann–Roch/effectiveness argument: if the reduced affine degree is below g the divisor is equivalent to an effective divisor, and conversely effectiveness forces the degree to drop under reduction. This is a genuine theorem, not a definition or a fitted input. Proposition 6 gives an iff criterion for MDS in terms of u-polynomial degrees; it is derived from the standard AG-code MDS condition (ℓ(G−D_I)=0 for all index sets of size t) plus Proposition 3, so checking those degrees is checking the classical condition, not renaming it. Theorem 4 computes the Euclidean dual explicitly from residues of η=dx/(p_D(x)(2y+h(x))) and uses the hypothesis u_A|2v_A+h to deduce G+H = div(u_A u_B)+(2g+2t−2)P∞; the resulting multiplier m is explicit and the LCD step follows from µ_P^2=m_P. Nothing is fitted and then 'predicted': the hypotheses are stated, and the examples verify them computationally in Magma. The self-citations ([14], [21], [22]) appear only as contextual references to prior constructions and are not load-bearing for the proofs here. The abstract's q=4 MDS LCD claim appears unsupported by the paper's own Example 2 (which establishes an LCP MDS [8,4,5] code, not an LCD code under Theorem 4's hypotheses), and the q=7 example reports [28,14] rather than [14,7]; these are correctness/consistency gaps, not circularity. No step in the derivation is equivalent by construction to its inputs, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

No new physical or mathematical entities are invented. The free parameters are the explicit Mumford polynomial choices and k, which are tuned to satisfy the hypotheses. The main ad hoc axioms are the 2-torsion condition, the square condition, and the exhaustive MDS degree conditions, all of which are stated but not proven for general q.

free parameters (2)
  • Mumford data (u_A,v_A,u_B,v_B) = Explicit polynomials in Examples 2-4 (e.g., q=5: u_A=x^2+2, v_A=0 and a degree-5 u_B)
    Hand-chosen or searched to satisfy gcd, square, and MDS conditions; no existence theorem is proven for all q, so the examples rest on these choices.
  • k (evaluation divisor index) = k=1 for q=4,5; k=2 for q=7
    Chosen to hit length/dimension targets; for q=7 the use of k=2 gives [28,14], not the abstract's [14,7].
assumptions (7)
  • standard math Riemann-Roch theorem: ℓ(G)=deg G+1-g+ℓ(K-G)
    Used throughout for non-speciality, code dimension, and the MDS criterion.
  • standard math Mumford representation: each divisor class in J(X) has a unique reduced pair (u,v) with deg u ≤ g
    Foundation for translating geometric non-speciality into a degree test; relies on Cantor reduction and the hyperelliptic model.
  • domain assumption Dimension formula for ℓ(D+sP∞) on hyperelliptic curves [16, Lemma 2.2.5]
    Loaded in the proof of Theorem 1; if the formula has hidden restrictions, the degree-g criterion could fail.
  • standard math Dual AG code formula C_L(D,G)^⊥=C_L(D,D+div(η)-G) with η=dt/t
    Used to compute the explicit dual in Theorem 4 and the multiplier m.
  • ad hoc to paper 2-torsion condition u_A | 2v_A + h
    Needed for 2D_A=div(u_A)+2gP∞ and for the dual to be a scaled C_L(D,H); the q=4 example does not satisfy it.
  • ad hoc to paper Square condition m_a ∈ (F_{q^2}^*)^2 for each evaluation x-coordinate a
    Required by Proposition 7 to construct μ with μ²=m; verified for the examples but not proven in general.
  • ad hoc to paper MDS theta-avoidance: for every t-subset I, the reduced u-degree of α±β-γ_I is g
    This is the hypothesis of Theorem 3; examples verify it computationally for q=4,5,7, but no general construction is proven.

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Pith. "Pith review of Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation." pith.science (2026). https://pith.science/paper/GSRYWRGD

@misc{pith2026260716945,
  author       = {Pith},
  title        = {Pith review of: Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GSRYWRGD}},
  note         = {Machine review of arXiv:2607.16945}
}
abstract

We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ with complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical hard geometric problem to a single-degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. Finally, we apply this framework to the maximal hyperelliptic curve $\mathcal{X} \colon y^2 = x^q + x$ over $\mathbb{F}_{q^2}$ and give explicit examples of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ for $q = 4, 5, 7$, verified computationally; we conjecture, with heuristic support, that such codes exist for all $q \geq 4$.

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