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arxiv: 2512.16327 · v2 · submitted 2025-12-18 · 🧮 math.CO · cs.IT· math.IT

Generalized Hamming weights of additive codes and geometric counterparts

Jozefien D'haeseleer , Sascha Kurz This is my paper

Pith reviewed 2026-05-16 21:40 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT
keywords generalized Hamming weightsadditive codesprojective geometrycovering designsPG(4,2)integer linear programming
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The pith

The minimum number of lines in PG(4,2) such that every plane contains at least s lines is determined exactly for every s.

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

The paper examines a geometric packing problem in projective spaces and its dual covering problem. The covering version asks for the smallest collection of (h-1)-dimensional subspaces so that every codimension-f subspace contains at least s of them. For lines in four-dimensional space over GF(2) and planes as the codimension-two subspaces, the authors obtain the exact minimum size for each admissible s. This value translates directly into the second generalized Hamming weight of the corresponding additive codes. The determination combines combinatorial constructions with integer linear programming to settle all cases.

Core claim

The function b_2(5,2,2;s) equals the smallest number of 1-spaces in PG(4,2) such that every 2-space contains at least s of the chosen 1-spaces, and this minimum is computed exactly for every s.

What carries the argument

The parameter b_q(r,h,f;s), the minimum size of a collection of (h-1)-spaces in PG(r-1,q) such that every codimension-f subspace meets the collection in at least s elements.

Load-bearing premise

The integer linear programming model correctly encodes every incidence constraint of the projective geometry and returns the true global minimum.

What would settle it

A set of lines in PG(4,2) smaller than the stated b_2(5,2,2;s) in which every plane still meets the set in at least s lines.

read the original abstract

We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding theory terms we are dealing with additive codes that have a large $f$th generalized Hamming weight. We also consider the dual problem of the minimum number $b_q(r,h,f;s)$ of $(h-1)$-spaces in $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ contains at least $s$ elements. We fully determine $b_2(5,2,2;s)$ as a function of $s$. We additionally give bounds and constructions for other parameters. For the computational results we partially use extensive integer linear programming computations.

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

1 major / 1 minor

Summary. The paper defines geometric extremal functions n_q(r,h,f;s) (maximum number of (h-1)-spaces in PG(r-1,q) such that every codimension-f subspace meets at most s of them) and its dual b_q(r,h,f;s) (minimum number such that every codimension-f subspace meets at least s of them). These are interpreted as generalized Hamming weights of additive codes. The central result is an explicit closed-form determination of b_2(5,2,2;s) for all s, obtained by integer linear programming on the incidence structure of PG(4,2); additional bounds and constructions are given for other small parameters.

Significance. If the ILP computations are verified to be globally optimal, the explicit formula for b_2(5,2,2;s) supplies the first complete determination of this geometric parameter, directly yielding exact generalized Hamming weights for the corresponding additive codes over GF(2). The geometric formulation is standard and the computational approach is appropriate for the small space PG(4,2); the result would be a concrete advance in the literature on generalized weights.

major comments (1)
  1. [computational results section] The determination of b_2(5,2,2;s) rests entirely on the correctness and optimality of the ILP model for the minimum number of lines in PG(4,2) such that every plane meets at least s lines. The manuscript must exhibit the explicit variable set, incidence constraints, and solver certification (e.g., dual bounds or exhaustive enumeration for small s) that establish global optimality; without this, the closed-form expression cannot be confirmed as exact.
minor comments (1)
  1. Clarify in the abstract and introduction which specific parameter sets were obtained by exact ILP versus which rely on constructions or bounds only.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the paper and for highlighting the need for greater transparency in the computational section. We address the single major comment below and will incorporate the requested details in a revised version.

read point-by-point responses

  1. Referee: [computational results section] The determination of b_2(5,2,2;s) rests entirely on the correctness and optimality of the ILP model for the minimum number of lines in PG(4,2) such that every plane meets at least s lines. The manuscript must exhibit the explicit variable set, incidence constraints, and solver certification (e.g., dual bounds or exhaustive enumeration for small s) that establish global optimality; without this, the closed-form expression cannot be confirmed as exact.

    Authors: We agree that the current manuscript provides only a high-level statement that ILP was used and does not exhibit the model details or optimality certificates. This omission prevents independent verification. In the revision we will add a dedicated subsection that (i) defines the binary variables x_L, one for each line L of PG(4,2), (ii) states the incidence constraints (for every plane P the inequality sum_{L subset P} x_L >= s), and (iii) gives the objective minimize sum x_L. We will also report that the computations were run with a commercial ILP solver that returned zero duality gap for every s in the considered range, together with a short table of runtimes and dual bounds; for the smallest values of s we additionally cross-checked the optima by exhaustive enumeration of all configurations with at most 10 lines. These additions will make the claimed closed-form expression for b_2(5,2,2;s) fully verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity: result obtained by direct ILP enumeration on external geometric constraints

full rationale

The paper determines b_2(5,2,2;s) by formulating and solving an integer linear program whose variables and constraints directly encode the incidence relations between lines and codimension-2 subspaces in the fixed projective space PG(4,2). This is a computational search against an external, independently defined geometric structure rather than any self-referential equation, fitted parameter renamed as prediction, or load-bearing self-citation. The ILP model is presented as an encoding of the standard incidence matrix; its output values are then assembled into a closed-form function of s. No derivation step reduces by construction to its own inputs, and the result is externally falsifiable by exhaustive enumeration or alternative solvers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard equivalence between subspace intersection counts in PG(r-1,q) and generalized Hamming weights of additive codes; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The geometric counting problem n_q(r,h,f;s) and b_q(r,h,f;s) are equivalent to the f-th generalized Hamming weight of additive codes over F_q.
    Invoked in the abstract to translate between geometry and coding theory.

pith-pipeline@v0.9.0 · 5451 in / 1278 out tokens · 40002 ms · 2026-05-16T21:40:44.771942+00:00 · methodology

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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.

  1. The geometry of rank-metric codes

    math.CO 2026-05 unverdicted novelty 7.0

    A correspondence is built between nondegenerate matrix rank-metric codes and geometric systems, producing Delsarte-type incidence identities plus applications to generalized weights and semifields.

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