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arxiv: 2604.16676 · v1 · submitted 2026-04-17 · 💻 cs.IT · cs.DM· math.AG· math.CO· math.IT· math.NT

Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes

Alain Couvreur , Rati Ludhani This is my paper

Pith reviewed 2026-05-10 06:53 UTC · model grok-4.3

classification 💻 cs.IT cs.DMmath.AGmath.COmath.ITmath.NT
keywords quadricsfinite fieldsprojective Reed-Muller codesminimal codewordsrational pointsabsolutely irreducible varieties
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The pith

Two absolutely irreducible quadrics over a finite field must coincide as varieties if one set of rational points contains the other, except one case over F2.

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

The paper establishes that absolutely irreducible quadrics over finite fields are uniquely determined by their rational points under inclusion: if the points of one lie inside those of the other, the two quadrics are identical except for a single exceptional pair over the field with two elements. This geometric fact directly classifies the minimal codewords of projective Reed-Muller codes of order 2, because those codewords correspond to quadrics whose rational-point sets are maximal under inclusion. Using the classification, the authors compute the exact number of minimal codewords of each possible weight. A reader cares because the result turns an open coding-theory question into a solved instance of point-set geometry over finite fields.

Core claim

Except for one particular case over F2, any two absolutely irreducible quadrics over a finite field whose sets of rational points are contained within one another must be equal as projective varieties. This yields a precise characterisation of the minimal codewords of projective Reed-Muller codes of order 2 together with their exact counts for each weight.

What carries the argument

The equivalence that identifies minimal codewords of projective Reed-Muller codes of order 2 with quadrics having maximal rational-point sets under inclusion, combined with the uniqueness theorem for absolutely irreducible quadrics.

If this is right

  • Minimal codewords of the projective Reed-Muller code of order 2 are in bijection with these maximal quadrics.
  • The exact number of minimal codewords is known for every admissible weight.
  • The classification holds over every finite field, with the single listed exception over F2 handled separately.

Where Pith is reading between the lines

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

  • The uniqueness result may simplify weight-distribution calculations for related algebraic codes.
  • Similar containment arguments could apply to higher-degree hypersurfaces whose rational points determine the variety.
  • The geometric criterion offers a test that could be implemented in computer-algebra systems to enumerate minimal codewords for small fields and dimensions.

Load-bearing premise

The quadrics are absolutely irreducible and the containment of rational points is taken with respect to the given finite field.

What would settle it

Exhibit two distinct absolutely irreducible quadrics over any finite field larger than F2 such that the rational points of one are properly contained in those of the other.

Figures

Figures reproduced from arXiv: 2604.16676 by Alain Couvreur, Rati Ludhani.

Figure 1
Figure 1. Figure 1: Reducible conics through P1, P2, P3, P4 with no three collinear. If q = 2, then there are 3 non collinear rational points on Q. Hence, the linear system interpolating these points is parameterized by P 2 : there are q 2 + q + 1 = 7 rational conics passing through these 3 points. Let us count the Fq-reducible conics again. Denote by P1, P2, P3 the 3 rational points and by ℓij the line passing through Pi and… view at source ↗
Figure 2
Figure 2. Figure 2: Reducible conics through non collinear points P1, P2, P3. Suppose Q ∼ E4 and Q′ ∼ E4. From Theorem 2.4, after a suitable linear change of variables, one can assume that F(X0, X1, X2, X3) = f(X0, X1) + X2X3 where f is Fq-irreducible. Then, Q ∩ V(X0) is parabolic of rank 3 and, from Proposition 2.19, Q′ ∩ V(X0) has rank 2 or 3. If Rk (Q′ ∩ V(X0)) = 2, then either it is a union of two rational lines or a unio… view at source ↗
read the original abstract

We study the classification of minimal codewords of projective Reed-Muller codes of order $2$. This problem is equivalent to identifying quadrics over finite fields whose set of rational points is maximal with respect to the inclusion. We prove that except one particular case over $\mathbb{F}_2$, any two absolutely irreducible quadrics whose sets of rational points are contained within one another should be equal as projective varieties. We deduce a precise characterisation of the minimal codewords of projective Reed-Muller codes of order $2$ and further give their exact number for each possible weight.

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 manuscript proves that, except for one explicit case over F_2, any two absolutely irreducible quadrics over a finite field whose F_q-rational point sets are nested under inclusion must coincide as projective varieties. This geometric statement is shown to be equivalent to the classification of minimal codewords in the projective Reed-Muller code of order 2; the authors then deduce an exact characterization of these codewords together with their number for each attainable weight.

Significance. The result supplies a parameter-free geometric criterion that directly yields the minimal supports and their multiplicities in PRM codes of order 2. The explicit handling of the single F_2 exception and the translation from quadric containment to codeword weight are strengths that make the classification complete and falsifiable.

minor comments (2)
  1. The abstract states the main theorem cleanly, but the precise definition of 'maximal with respect to the inclusion' (i.e., maximality of the point set) should be restated once in the introduction for readers who enter via the coding-theoretic side.
  2. In the application to Reed-Muller codes, the correspondence between the support of a codeword and the zero set of a quadratic form is used without an explicit small-field example; adding one (e.g., q=3 or q=4) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, including the recognition of the geometric criterion for nested rational point sets on absolutely irreducible quadrics and the explicit treatment of the single F_2 exception. We appreciate the assessment that this yields a complete classification of minimal codewords in projective Reed-Muller codes of order 2.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper equates the classification of minimal codewords in projective Reed-Muller codes of order 2 with the identification of quadrics maximal under inclusion of F_q-rational points. It then proves directly, via case analysis on quadratic forms and handling of the single F_2 exception, that absolutely irreducible quadrics are uniquely determined by their point sets under inclusion. This uniqueness is established from first principles in algebraic geometry over finite fields and is not obtained by fitting parameters, self-definition, or load-bearing self-citations; the codeword characterization follows immediately as a translation of the geometric result without circular reduction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about absolutely irreducible quadrics and their rational points over finite fields; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Absolutely irreducible quadrics over finite fields have well-defined rational point sets that behave under inclusion.
    Invoked to equate nested point sets with equality of varieties.

pith-pipeline@v0.9.0 · 5400 in / 1221 out tokens · 41331 ms · 2026-05-10T06:53:07.523890+00:00 · methodology

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Reference graph

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