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arxiv: 2311.08992 · v2 · submitted 2023-11-15 · 💻 cs.IT · math.IT· math.NT

Lifting iso-dual algebraic geometry codes

Mar\'ia Chara , Ricardo Podest\'a , Luciane Quoos , Ricardo Toledano This is my paper

Pith reviewed 2026-05-24 06:19 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.NT
keywords iso-dual codesalgebraic geometry codesfunction field extensionsseparable extensionscode liftingcyclotomic extensionsHermitian function field
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The pith

A method lifts an iso-dual AG-code over a function field F to an iso-dual code over a separable extension M when divisors D and G meet stated conditions and different exponents have suitable parity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general lifting technique that takes an existing iso-dual algebraic geometry code defined over a base function field and produces a new iso-dual code over a finite separable extension of that field. The construction proceeds by extending the divisors that define the original code and checking that the ramification data in the extension preserves the iso-dual property through parity conditions on the different exponents. The authors then apply the method to concrete cases, starting from codes over the rational function field and obtaining new codes over elementary abelian p-extensions, including maximal function fields, as well as long binary and ternary codes over cyclotomic extensions. A reader would care because the technique supplies a systematic route to iso-dual codes of greater length or over larger alphabets without having to construct them from scratch in each new field.

Core claim

Given a finite separable extension M/F of function fields and an iso-dual AG-code C defined over F, there is a general method to lift C to another iso-dual AG-code tilde C defined over M whenever the divisors D and G and the parity of the involved different exponents satisfy the required technical assumptions.

What carries the argument

The lifting construction that extends the code C to tilde C over M by modifying the divisors D and G so that the parity conditions on different exponents keep the dual equal to the code itself.

If this is right

  • Iso-dual AG-codes over the rational function field lift to elementary abelian p-extensions.
  • The method yields iso-dual codes over the Hermitian, Suzuki, and GGS maximal function fields.
  • Long binary and ternary iso-dual AG-codes are obtained over cyclotomic extensions.
  • The lifting applies to any pair of function fields and divisors satisfying the parity and divisor hypotheses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting idea might generate iso-dual codes of record length in fields where direct constructions are difficult.
  • Performance comparisons between lifted codes and independently constructed codes over the same extension could test whether the method improves parameters.
  • The parity condition on different exponents may suggest analogous lifting results for other classes of codes that rely on duality.

Load-bearing premise

The chosen divisors D and G together with the parity of the different exponents in the extension M/F must satisfy the technical conditions that keep the lifted code iso-dual.

What would settle it

An explicit separable extension M/F, divisors D and G, and parity data that meet all stated conditions yet produce a lifted code whose dual is not equal to itself would show the method does not work.

Figures

Figures reproduced from arXiv: 2311.08992 by Luciane Quoos, Mar\'ia Chara, Ricardo Podest\'a, Ricardo Toledano.

Figure 1
Figure 1. Figure 1: Decomposition of places of Fq 2 (x) in M. Now we compute the number of Fq 2 -rational points on the curve X . The place at infinity R∞ is rational. Clearly, there are q places of the form Rα,0 with α ∈ S0. Let R γ α,β be a rational place in M over Qα,β in F for β 6= 0, then γ q+1 = β q + β and β q+1 = α q + α 6= 0. We consider two cases. Case 1 : If β q + β = 0, then β q+1 = α q + α implies −β 2 = α q + α,… view at source ↗
Figure 2
Figure 2. Figure 2: Decompositions of Pf and P∞. We will define a cyclic subgroup H of (R/(x n ))∗ of order q m generated by a suitable residual class x − α for some α ∈ Fq. With this choice of H we have that the subfield Kn = F(Λxn ) H defines a cyclic extension F(Λxn)/Kn of degree q m and we will try to lift a rational iso-dual AG-code to an iso-dual AG-code over Kn. For an estimate of the minimum distance of these liftings… view at source ↗
Figure 3
Figure 3. Figure 3: Ramification of Sx. (b) Assume that F(Λx) ⊂ Kn. From the ramification situation described in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
read the original abstract

In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an iso-dual AG-code $\mathcal{C}$ defined over $\mathcal{F}$, we provide a general method to lift the code $\mathcal{C}$ to another iso-dual AG-code $\tilde{\mathcal{C}}$ defined over $\mathcal{M}$ under some assumptions on the divisors $D$ and $G$ and on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian $p$-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the $GGS$ function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that given a finite separable extension M/F of function fields and an iso-dual AG-code C over F, there exists a general lifting method producing an iso-dual AG-code over M, provided assumptions on the divisors D and G and on the parity of the different exponents hold. The method is applied to lift iso-dual codes from the rational function field to elementary abelian p-extensions (Hermitian, Suzuki, GGS) and to construct long binary and ternary iso-dual AG-codes over cyclotomic extensions.

Significance. If the lifting construction is valid under the stated hypotheses, the work supplies a systematic tool for extending iso-dual AG codes to larger fields from base-field examples. The concrete applications to maximal function fields and cyclotomic extensions yield explicit families of iso-dual codes over small alphabets, which may be useful for further theoretical or computational work in algebraic coding theory.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'under some assumptions' is imprecise; a brief enumeration or forward reference to the precise conditions on D, G, and different-exponent parities (as stated in the main theorem) would improve readability.
  2. Notation for the lifted code and divisors should be introduced consistently in the first section where the lifting is defined, to avoid later ambiguity when comparing C and tilde C.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its potential utility in algebraic coding theory, and the recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a conditional lifting construction for iso-dual AG-codes from a base field F to an extension M, relying on explicit assumptions about divisors D, G and different exponent parities. This is a direct algebraic construction from function-field properties, not a fitted parameter renamed as prediction, not a self-definition, and not dependent on load-bearing self-citations or imported uniqueness theorems. The applications to Hermitian, Suzuki, GGS, and cyclotomic cases are presented as instances satisfying the stated hypotheses rather than as derivations that collapse to the inputs. The derivation chain is self-contained against external benchmarks and does not reduce by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract alone; full list of free parameters, axioms, and invented entities cannot be extracted. The method relies on standard background results from algebraic function fields and algebraic-geometry codes.

axioms (1)
  • standard math Standard properties of algebraic geometry codes and ramification in separable extensions of function fields
    Invoked implicitly as the setting in which the lifting is defined.

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Reference graph

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