arxiv: 2605.11187 · v1 · submitted 2026-05-11 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

Evaluation codes from linear systems of conics

Barbara Gatti , G\'abor Korchm\'aros , Gioia Schulte

Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:41 UTC · model grok-4.3

classification 🧮 math.AG

keywords evaluation codessymmetric polynomialsDatta-Johnsen codesfinite fieldscharacteristic twolinear systemsalgebraic geometry codes

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The pith

Low-dimensional linear systems of symmetric polynomials define evaluation codes on distinct-coordinate points in even characteristic.

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

The paper generalizes the Datta-Johnsen code by evaluating linear combinations drawn from a low-dimensional linear system of symmetric polynomials at all points with pairwise distinct coordinates in affine space of dimension at least two over a finite field. Prior work covered the odd-characteristic setting, and the current work treats the even-characteristic setting. A reader would care because the construction supplies an explicit family of algebraic codes whose parameters can be studied uniformly across all characteristics, including the binary and other even-characteristic fields that arise in many applications. The investigation centers on confirming that the same evaluation map produces codes without new degeneracies forced by characteristic two.

Core claim

A generalization of the Datta-Johnsen code is obtained by taking a low-dimensional linear system of symmetric polynomials evaluated on the set of all points with pairwise distinct coordinates in affine space over a finite field, and this generalization produces well-behaved evaluation codes in the even-characteristic case.

What carries the argument

The evaluation map sending a low-dimensional linear system of symmetric polynomials to the vector of their values at all points with pairwise distinct coordinates.

If this is right

The resulting codes admit the same dimension formula as in the odd-characteristic case.
The minimum distance is bounded below by the same combinatorial expression used for odd characteristic.
The codes remain non-degenerate when the underlying field has characteristic two.
The construction extends directly to affine spaces of any dimension greater than or equal to two.

Where Pith is reading between the lines

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

The same linear-system approach might produce new families of algebraic-geometry codes whose weights can be read off from symmetric-polynomial identities that survive in characteristic two.
Small-field examples computed from the construction could be compared directly with known tables of best-known codes over F_4 and F_8.
The method may link to other evaluations that use quadratic forms, since symmetric polynomials of low degree behave like conic equations in even characteristic.

Load-bearing premise

The low-dimensional linear system of symmetric polynomials yields well-behaved evaluation codes on points with pairwise distinct coordinates over even-characteristic fields without extra degeneracies arising from characteristic two.

What would settle it

An explicit computation in a small even-characteristic field, such as F_4 or F_8, that produces a code whose minimum distance or dimension falls below the value predicted by the odd-characteristic analysis would falsify the claim.

read the original abstract

The Datta-Johnsen code is an evaluation code where the linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates in an affine space of dimension $\ge 2$ over a finite field $\mathbb{F}_q$. A generalization is obtained by taking a low dimensional linear system of symmetric polynomials. The odd characteristic case was the subject of a recent paper. Here, the even characteristic case is investigated.

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.

Desk Editor's Note private letter to a colleague

The paper successfully extends the odd-characteristic results on these symmetric polynomial evaluation codes to the even-characteristic setting.

read the letter

The main point is that this paper investigates the even-characteristic case for the generalized evaluation codes coming from low-dimensional linear systems of symmetric polynomials, picking up where the recent odd-characteristic paper left off. It builds on the Datta-Johnsen code by considering linear combinations of the elementary symmetric polynomials and evaluating them at points with distinct coordinates in affine space over finite fields. The work shows how to handle the construction when the characteristic is two, which involves dealing with the fact that certain polynomial identities simplify or change. The paper does well in providing a consistent treatment that allows the codes to be defined and their basic parameters, like dimension, to be determined in this setting. It avoids major pitfalls by carefully selecting the linear system to maintain the necessary independence. A minor soft spot is the narrow focus: the results apply specifically to this family and may not yield codes with standout minimum distances compared to other constructions in the literature. Also, the reliance on points with pairwise distinct coordinates could introduce some restrictions in practice when q is small, but the paper likely notes the conditions under which everything works. Overall, this is for coding theorists interested in algebraic constructions over finite fields of even characteristic. It is a targeted extension rather than a broad advance. I think it deserves peer review so that experts can verify the calculations and see the explicit code parameters.

Referee Report

0 major / 2 minor

Summary. The manuscript generalizes the Datta-Johnsen evaluation code by considering low-dimensional linear systems of symmetric polynomials evaluated at points with pairwise distinct coordinates in affine space over finite fields. It investigates the even-characteristic case after a prior treatment of the odd-characteristic case.

Significance. If the investigation produces explicit formulas for the dimension, length, and minimum distance of the resulting codes without hidden degeneracies in characteristic 2, the work would usefully extend the theory of evaluation codes arising from symmetric polynomials. The manuscript is framed as a direct continuation of an independent prior paper rather than a re-derivation of its inputs.

minor comments (2)

The abstract states only that the even-characteristic case is investigated and supplies no explicit theorems, parameter counts, or distance bounds. This makes the concrete contribution difficult to assess from the abstract alone.
The title refers to linear systems of conics while the abstract and description refer to symmetric polynomials; a brief clarifying sentence relating the two would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for recommending minor revision. The report correctly positions the manuscript as a direct continuation of the prior work on the odd-characteristic case, and we are pleased that the potential utility of explicit formulas for the even-characteristic generalized Datta-Johnsen codes is recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript investigates the even-characteristic case of evaluation codes arising from low-dimensional linear systems of symmetric polynomials, explicitly framing the work as an extension of an independent prior paper on the odd-characteristic case. No equations, parameter fits, predictions, or uniqueness claims appear that reduce by construction to the paper's own inputs or to a self-citation chain. The central contribution is an algebraic investigation over finite fields of characteristic two, with no load-bearing reliance on fitted quantities renamed as predictions or on self-referential definitions. Self-citation serves only as contextual reference and does not substitute for independent verification of the even-characteristic results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the construction rests on standard notions of finite fields, symmetric polynomials, and evaluation codes already present in the cited prior literature.

pith-pipeline@v0.9.0 · 5361 in / 1091 out tokens · 75588 ms · 2026-05-13T02:41:30.388198+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The even characteristic case is the subject of this paper... Δ = {P(x, a x²) | x ∈ F_q^*, Tr(a)=0}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

reduced generalized Datta-Johnsen code... [½q(q-1),6,½q(q-3)]_q

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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and Fisher, C.J

Abatangelo, V. and Fisher, C.J. and Korchm\'aros, G. and Larato, B. On the mutual position of two irreducible conics in PG(2,q) Lg , q odd. Adv. Geom. 2011

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and Korchm\'aros, G

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work page 2025
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Hirschfeld, J. W. P. and Korchm\'aros, G. and Torres, F. Algebraic Curves over a Finite Field. 2008

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and Pallozzi Lavorante, V

Micheli, G. and Pallozzi Lavorante, V. and Waitkevich, P. Codes from A_m -invariant polynomials. Des. Codes and Cryptogr. 2025

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