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Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions

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Pith's one-line read An arithmetic characterization of non-special divisors on Kummer extensions permits explicit constructions of degree g-1 divisors on the GK curve and a framework for LCP and LCD algebraic geometry codes.

desk verdict The paper's main advance is an arithmetic test for non-special divisors on Kummer extensions that works when the support includes non-totally ramified places, plus explicit constructions and an LCP framework that follows from it. read the letter →

arxiv 2606.11764 v1 pith:V2RT4QHP submitted 2026-06-10 cs.IT math.IT

Huachao Zhang , Chang-An Zhao This is my paper · ORCID
classification cs.ITmath.IT
keywords non-specialdivisorsKummerextensionsGKcurvelinearcomplementarypairsLCDcodesalgebraicgeometrypuregapsfunctionfields
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 establishes an arithmetic characterization of non-special divisors on Kummer extensions y^m = f(x) that remains valid when the support includes non-totally ramified places. Using this, the authors construct explicit non-special divisors of degree g-1 on the GK curve. They also give families of effective non-special divisors of degree g via pure gaps with uniform multiplicities. These tools support a general framework for linear complementary pairs of algebraic geometry codes on Kummer extensions, where canonical divisors determine security parameters and enable LCD code constructions, illustrated by examples on the GK curve and Hermitian quotients.

What carries the argument

The arithmetic characterization of non-special divisors whose support can contain non-totally ramified places, which enables the explicit constructions and the LCP framework.

What would settle it

An explicit divisor on the GK curve whose support includes a non-totally ramified place but fails to satisfy the claimed non-special property, or a constructed LCP whose security parameters cannot be recovered from the canonical divisor as stated.

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Extended reading notes

Core claim

The paper claims that non-special divisors on Kummer extensions admit an arithmetic characterization allowing non-totally ramified places in the support; this yields explicit degree g-1 constructions on the GK curve, uniform-multiplicity degree g families via pure gaps, a framework for LCPs of AG codes whose security parameters follow from canonical divisors, and constructions of LCD AG codes.

Load-bearing premise

The arithmetic characterization of non-special divisors remains valid when the support includes places that are not totally ramified.

Editorial extensions

If this is right

  • Explicit non-special divisors of degree g-1 exist on the GK curve.
  • Several families of effective non-special divisors of degree g with identical multiplicities exist on general Kummer extensions.
  • A general framework exists for constructing LCPs of AG codes on Kummer extensions.
  • Security parameters of such LCPs are determined by canonical divisors within the framework.
  • LCD AG codes can be constructed on the same Kummer extensions.

Reading between the lines

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

  • The same arithmetic test might classify non-special divisors on other classes of curves that admit Kummer-like descriptions.
  • The uniform-multiplicity families could be combined with different place supports to produce codes with varying minimum distances.
  • The LCP framework might extend to towers of Kummer extensions by iterating the canonical-divisor argument.
  • Explicit LCD constructions on Hermitian quotients suggest a pattern that could apply to other maximal curves.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes an arithmetic characterization of non-special divisors on Kummer extensions y^m = f(x) whose support may include non-totally ramified places. It applies this to give explicit constructions of non-special divisors of degree g-1 on the GK curve, families of effective degree-g non-special divisors via pure gaps on general Kummer extensions, a general framework for LCP AG codes on these extensions, and a method to determine security parameters of the LCPs from canonical divisors (also yielding LCD codes). The results are illustrated with examples on the GK curve and quotients of the Hermitian curve.

Significance. If the characterization holds, the work supplies explicit, usable constructions together with a parameter-determination method for LCP and LCD AG codes; this is a concrete advance for the construction of complementary codes from function fields.

minor comments (2)
  1. The abstract refers to 'representative examples' without listing the specific curves, degrees, or code parameters; adding a short table or explicit parameter list in the introduction would improve readability.
  2. Notation for the places and ramification in the Kummer extension is introduced in the abstract but would benefit from a dedicated preliminary subsection with consistent symbols before the characterization is stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript, including the recommendation for minor revision. The report contains no enumerated major comments requiring point-by-point responses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper derives an arithmetic characterization of non-special divisors on Kummer extensions y^m = f(x) that explicitly allows non-totally ramified places, then applies it to explicit constructions of degree g-1 divisors on the GK curve and families using pure gaps. These steps rest on standard function-field machinery (Riemann-Roch via canonical divisors) without reducing any claimed result to a fitted parameter, self-citation chain, or definitional renaming inside the paper. The LCP/LCD framework follows directly from the characterization and is externally falsifiable via code parameters. No load-bearing step matches the enumerated circularity patterns.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on the standard theory of algebraic function fields and places; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

assumptions (1)
  • standard math Standard properties of ramification and divisors on algebraic function fields over finite fields hold for the Kummer extensions y^m = f(x).
    Invoked when the authors define the curves and speak of totally ramified versus non-totally ramified places.

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Cite this review

Pith. "Pith review of Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions." pith.science (2026). https://pith.science/paper/V2RT4QHP

@misc{pith2026260611764,
  author       = {Pith},
  title        = {Pith review of: Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2RT4QHP}},
  note         = {Machine review of arXiv:2606.11764}
}
abstract

Recently, constructions of linear complementary pairs (LCPs) of codes and linear complementary dual (LCD) codes on function fields have attracted considerable attention due to the wide range of applications of these codes. Such constructions rely on non-special divisors of degrees $g$ and $g-1$. In this work, we investigate Kummer extensions defined by $y^m = f(x)$ with $f(x)\in\mathbb{F}_q(x)$ and establish an arithmetic characterization of non-special divisors whose support can contain non-totally ramified places. Based on this characterization, we explicitly construct non-special divisors of degree $g-1$ on the GK curve. Moreover, utilizing pure gaps, we explicitly provide several families of effective non-special divisors of degree $g$ on Kummer extensions with the same multiplicities. We then develop a general framework for constructing LCPs of algebraic geometry (AG) codes on Kummer extensions. By virtue of canonical divisors, we show that the security parameters of LCPs of AG codes can be determined within this framework, which also enables the construction of LCD AG codes. Finally, we illustrate our results with representative examples, including LCPs of codes on the GK curve and LCD codes on quotients of the Hermitian curve.

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