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Linear Complementary Pairs of Algebraic Geometry Codes via Kummer Extensions

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Pith's one-line read This paper constructs explicit non-special divisors on Kummer extensions and uses them to build linear complementary pairs (LCPs) of algebraic geometry codes, including MDS LCPs from elliptic function fields and LCD codes from maximal…

desk verdict Central Theorem 5 rests on a false floor/ceiling identity, and the examples contradict the paper's own S_t definition; the LCP families are unsupported as written. read the letter →

arxiv 2506.23081 v1 pith:4RP4PG2W submitted 2025-06-29 cs.IT math.IT

Huang Junjie , Chen Haojie , Zhang Huachao , Zhao Chang-An This is my paper · ORCID
classification cs.ITmath.IT MSC 94B2711T7114H5514G50
keywords linearcomplementarypairsalgebraicgeometrycodesKummerextensionsnon-specialdivisorsWeierstrasssemigroupsMDSLCDellipticfunctionfields
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 aims to make Kummer extensions—function fields of the form $y^m = a\prod_i(x-\alpha_i)\prod_j(x-\beta_j)^{\lambda_j}\prod_k(x-\gamma_k)^{\lambda'_k}$—into a source of linear complementary pairs (LCPs) of algebraic geometry codes. The central move is an explicit divisor $A$ of degree $g$ (and, by removing a rational place, of degree $g-1$) that is effective and non-special, built from Weierstrass semigroup data. With such divisors in hand, the paper invokes a standard sufficient condition for LCPs of AG codes and produces families whose security parameter is the minimum distance of one constituent code. The applications cover the maximal subcovers $X_m$ and $Y_m$, elliptic function fields, hyperelliptic function fields, and further Kummer families; a corollary yields MDS LCPs over elliptic fields, and the final section yields LCD codes from maximal elliptic and hyperelliptic fields. If correct, the paper gives many explicit, tunable LCP families with known distance bounds.

What carries the argument

The central object is the explicit non-special divisor $A$ of degree $g$, assembled from the Weierstrass semigroup gaps of the places $Q_i$ and $Q'_j$ in the Kummer extension. The identity doing the work is the difference sequence $s_t=\ell_t-\ell_{t+1}$ for $1\le t\le m-2$, with $s_{m-1}=\ell_{m-1}-1$, where $\ell_t=S_t$ counts the residue classes of the form $m\mu+t$ that are gaps; the proof that $\deg A=g$ is a floor/ceiling summation using the classical identity for sums of floors. The conversion mechanism is the LCP sufficient condition: if $\gcd(G,H)$ and $\operatorname{lmd}(G,H)-D$ are non-special, then $(C_L(D,G),C_L(D,H))$ is an LCP, and the paper chooses $G,H$ so that these two divisors are exactly the constructed $A$ and $A-P$.

What would settle it

Compute $S_t$ and $s_t$ exactly as defined in Section III for the paper's Example 10(1), with $q=5$, $m=7$, and $y^7=(x-2)(x-3)(x-4)x^6(x-1)^6$. If direct substitution yields $s_6=-1$, then the divisor $A$ of Theorem 5 is not effective, and the LCP families built on it lack their stated foundation.

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Extended reading notes

Core claim

On its own terms, the central claim is an explicit recipe for non-special divisors on Kummer extensions with mixed exponents (Equation (4)). Under the conditions $0\le v,w\le\lfloor\lambda_0/m\rfloor$, $\lambda_j\in V_F$, and $s_t\ge 0$, the divisor $A=\sum_{t=1}^{m-2} t\sum_{\ell=1}^{s_t}Q_{t\ell}+(m-1)\sum_{\ell'=1}^{s_{m-1}}P_{\ell'}$ is effective and non-special of degree $g$. Theorems 11, 13 and 15 convert this divisor into LCP pairs $(C_L(D,G),C_L(D,H))$ with explicit lengths, dimensions, and distance lower bounds, and they prove the duality relation $C_L(D,H)^\perp\sim C_L(D,G)$, so the security parameter is $d(C_L(D,G))$; in the elliptic case, Corollary 22 makes these pairs MDS. The paper also constructs LCD codes from maximal elliptic and hyperelliptic fields by choosing $H$ so that $C_L(D,G)^\perp=C_L(D,H)$.

Load-bearing premise

Everything rests on the claim that the explicit divisor $A$ defined by the gap-count sequence is effective and non-special; that claim needs the arithmetic identity $s_{m-1}=\lceil\lambda_0/m\rceil\ge0$ to hold, and the proof asserts this identity rather than deriving it from the preceding floor/ceiling algebra.

Editorial extensions

If this is right

  • Every Kummer extension satisfying the arithmetic conditions in Theorem 5 supplies an explicit non-special divisor of degree $g$, hence also one of degree $g-1$ for every rational place outside its support.
  • Theorems 11, 13 and 15 turn each such divisor into an LCP of AG codes whose two constituents have complementary dimensions and whose security parameter is explicitly $d(C_L(D,G))$.
  • The construction covers Kummer extensions with $m<\lambda_0$ and with mixed ramified and unramified places, going beyond earlier treatments that required $m>\lambda_0$.
  • Over elliptic function fields, Corollary 22 produces MDS LCPs, so the security parameter reaches the Singleton bound; over maximal elliptic and hyperelliptic fields, Theorem 24 produces LCD codes.
  • On the maximal subcovers $X_m$ and $Y_m$, the construction gives explicit long LCPs over $\mathbb{F}_{q^6}$ and $\mathbb{F}_{q^{2r}}$ with stated distance lower bounds.

Reading between the lines

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

  • A direct check the paper does not perform is a computer search over exponent tuples $(u,v,w,\lambda_j,\lambda'_k,m)$ to map exactly which Kummer extensions satisfy the hypotheses of Theorem 5.
  • If the MDS LCP corollary is correct, these codes are natural explicit ingredients for direct-sum masking schemes, since both the fault-injection distance and the side-channel distance are fixed by the construction; the paper stops at the code parameters.
  • The same gap-count machinery could in principle be applied to other Galois covers of $\mathbb{F}_q(x)$ whose Weierstrass semigroups are known, not just Kummer extensions.
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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

4 major / 2 minor

Summary. The manuscript proposes explicit constructions of non-special divisors of degree g and g-1 on Kummer extensions of the form y^m = a * prod (x-alpha_i) * prod (x-beta_j)^{lambda_j} * prod (x-gamma_k)^{lambda'_k}, with partial ramification exponents. It then uses these divisors, through the Bhowmick-Dalai-Mesnager criterion, to construct linear complementary pairs (LCPs) of algebraic geometry codes, LCD codes, and MDS LCPs from subcovers of the BM curve, elliptic function fields, hyperelliptic function fields, and other function fields. The main technical engine is Theorem 5, which asserts that a divisor A assembled from the coefficients s_t = S_t - S_{t+1} and s_{m-1} = S_{m-1} - 1 is effective, non-special, and of degree g under the conditions 0 <= v,w <= floor(lambda_0/m), lambda_j in VF, and s_t >= 0. Theorems 11, 13, 15, 17, 19, 21, and 26, together with Corollary 22, convert this construction into families of LCP/LCD codes with stated parameters and security parameters.

Significance. If Theorem 5 were correct, the paper would provide useful explicit families of LCPs of AG codes, including MDS LCPs from elliptic function fields, and would extend earlier work of Moreno-Lopez-Matthews and Castellanos et al. by allowing more general ramification exponents. The paper is organized around a clear strategy and makes use of established external results on Weierstrass semigroups; I found no circularity or parameter fitting. However, Theorem 5 is the single load-bearing point, and the manuscript's own examples contradict the definitions on which that theorem rests. As written, the claimed LCP and LCD parameter families are not established.

major comments (4)
  1. [Section III, Theorem 5] The computation of s_{m-1} in the proof of Theorem 5 is algebraically incorrect. Since floor((m-1)a/m) = a - ceil(a/m), the definition of S_t gives S_{m-1} = u + sum_j (lambda_j - ceil(lambda_j/m)) + sum_k (lambda'_k - ceil(lambda'_k/m)) - (lambda_0 - ceil(lambda_0/m)) = ceil(lambda_0/m) - v - w, and therefore s_{m-1} = ceil(lambda_0/m) - v - w - 1, not ceil(lambda_0/m). The manuscript drops the contributions of ceil(lambda_j/m) and ceil(lambda'_k/m). This invalidates the claimed monotonicity of the sequence ell_t and the well-definedness of A. The contradiction is visible in Example 10(1): for u=3, v=2, lambda_1=lambda_2=6, m=7, lambda_0=15, the definition S_t = 3 + 2 floor(6t/7) - floor(15t/7) gives S_t = 1 for every 1 <= t <= 6, hence s_6 = 0, not the value implicit in the proof.
  2. [Section III, Theorem 5] The degree computation in the proof of Theorem 5 is also incorrect. Summing S_t directly and applying Lemma 8(iii) gives deg A = (m-1)u + sum_j ((lambda_j-1)(m-1))/2 + sum_k (((lambda'_k-1)(m-1) + gcd(lambda'_k,m) - 1))/2 - ((lambda_0-1)(m-1))/2, whereas the proof inserts an extra factor (m-1) inside the sums over j and k. For Example 10(1), the direct computation gives sum_{t=1}^6 S_t = 6 and s_t = 0 for all t, so deg A = 0 rather than g = 12. Thus the assertion that A has degree g is numerically contradicted by the paper's own definitions.
  3. [Section III, VF definition and Example 10] Example 10(1) also contradicts the definition of VF. For lambda_j = 6 and m = 7, the inverse lambda is 6, and the formula for f_j(t) gives f_j(1) = 3 floor(6/7) + 2 floor(36/7) - floor(90/7) - 1 = -3. Hence 6 is not in VF under the manuscript's own definition, so the hypothesis lambda_j in VF in Theorem 5 is not satisfied by the paper's leading example. The same type of inconsistency appears in Examples 10(2) and 10(3): in Example 10(2), S_1 = 2 and S_5 = 0, not ell_1 = 4 and ell_5 = 2, and in Example 10(3), S_1 = 2, not ell_1 = 3.
  4. [Sections IV-V, dependence on Theorem 5] Theorems 11, 13, 15, 17, 19, 21, and 26 and Corollary 22 all invoke Theorem 5 to obtain the non-special divisors required by Lemma 4. Since the proof of Theorem 5 is incorrect and the examples contradict the definitions, none of these LCP/LCD parameter statements is currently supported. In particular, the MDS optimality claims in Corollary 22 depend on the same construction and are therefore also unsupported.
minor comments (2)
  1. [Section III, notation] The symbol lambda is used both for the exponents lambda_j and for the multiplicative inverse modulo m, which makes the formulas in Section III difficult to verify.
  2. [Theorems 17 and 19] The conditions VF = {lambda, d'} and VF = {d'} are asserted without a general proof or verification procedure except in the worked numerical examples; since these conditions are load-bearing for applying Theorem 5, the manuscript should either prove them in general or state them as explicit hypotheses with a method of checking them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-special divisor construction and LCP theorems are derived from external semigroup/Riemann-Roch results, with no fitted parameters or self-citation chains.

full rationale

The derivation chain is self-contained against external results: Theorem 5 rests on Lemmas 6-8 from Refs. [26], [32], [33] (Weierstrass semigroup descriptions and floor-sum identities), and the LCP conversion rests on Lemma 4 from Ref. [25] and Lemma 3 from Ref. [29]. None of these are cited to the present authors, so there is no self-citation chain. The parameters s_t and VF are defined arithmetically and then hypothesized nonnegative; Theorem 5 proves effectiveness and non-specialness from those hypotheses rather than fitting them to the conclusion. The MDS argument in Corollary 22 uses a standard Jacobian group law and is independent of the Kummer construction. I note for completeness that the proof of Theorem 5 contains an apparent arithmetic slip: after writing floor((m-1)a/m) = a - ceil(a/m), the sum with u + sum lambda_j + sum lambda'_k = lambda0 yields s_{m-1} = floor(lambda0/m) - v - w, not ceil(lambda0/m), and Example 10(1) appears to conflict with the stated S_t table; however this is a correctness/verification concern, not circularity. Nothing in the paper reduces a predicted quantity to a fitted input or imports a uniqueness theorem from the same authors.

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

The central claim rests on standard algebraic function field theory plus external Weierstrass semigroup descriptions. No numerical fitting is present. Instead, several ad hoc positivity conditions (VF, s_t>=0) are asserted in applications and are internally inconsistent in the examples, which is the main source of correctness risk.

assumptions (5)
  • standard math Riemann-Roch theorem and standard divisor theory for function fields
    Used throughout Section II and Section III to compute dimensions, non-special divisors, and code parameters.
  • standard math Kummer extension genus and different formulas from Stichtenoth
    Used in Theorem 11, Theorem 15, and Theorem 24 to compute genus, different divisors, and differential divisors.
  • standard math Weierstrass semigroup descriptions from [32, Prop. 3.12] and [33, Cor. 3.5]
    Lemma 6 and Lemma 7 are cited without proof; they give the gap descriptions used in Theorem 5.
  • domain assumption Existence of rational places a_i that split completely in the Kummer extension
    Theorems 11, 13, 15, and 17 assume a divisor D formed from such split places; in applications D is taken as all rational places outside the ramified locus.
  • ad hoc to paper Conditions VF = {...} and s_t >= 0 hold in each application
    Theorem 5 requires these conditions, but the examples assert them with values that do not match the paper's own formulas, e.g., Example 10(1) and Example 10(2).

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Pith. "Pith review of Linear Complementary Pairs of Algebraic Geometry Codes via Kummer Extensions." pith.science (2026). https://pith.science/paper/4RP4PG2W

@misc{pith2026250623081,
  author       = {Pith},
  title        = {Pith review of: Linear Complementary Pairs of Algebraic Geometry Codes via Kummer Extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RP4PG2W}},
  note         = {Machine review of arXiv:2506.23081}
}
abstract

Due to their widespread applications, linear complementary pairs (LCPs) have attracted much attention in recent years. In this paper, we determine explicit construction of non-special divisors of degree $g$ and $g-1$ on Kummer extensions with specific properties. In addition, we present several methods for constructing LCPs of algebraic geometry codes (AG Codes) via Kummer extensions. These results are applied in constructing explicit LCPs of AG Codes from subcovers of the BM curve, elliptic function fields, hyperelliptic function fields and other function fields. It is important to mention that we construct several families LCPs of MDS AG Codes from elliptic function fields and we obtain some linear complementary dual (LCD) codes from certain maximal elliptic function fields and hyperelliptic function fields.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.AG 2026-07 conditional novelty 7.0 of 10

    A degree test on Mumford polynomials characterizes non-special divisors on hyperelliptic curves and yields explicit LCP/LCD code constructions, with a verified q=5 MDS LCD example.

Reference graph

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