arxiv: 2605.02533 · v1 · submitted 2026-05-04 · 🧮 math.CO · math.GR· math.RA

Recognition: 3 theorem links

· Lean Theorem

Self-dual codes with group actions and invariants

Futo Takabayashi

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:05 UTC · model grok-4.3

classification 🧮 math.CO math.GRmath.RA

keywords self-dual codesgroup actionsweight enumeratorsMacWilliams identityClifford-Weil groupsGleason theoremsfinite ringsbilinear forms

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The pith

The G-full weight enumerators of G-self-dual and G-isotropic codes are invariant under Clifford-Weil groups and span the invariant subspaces.

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

The paper defines dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and generalizes Hayden's theorem to this setting. It introduces G-codes, which are codes invariant under a permutation group G, along with G-quadratic maps and G-representations. Generalized MacWilliams identities are established for G-codes and for G-full weight enumerators. The central extension is a G-analogue of Gleason-type theorems showing that the G-full weight enumerators of G-self-dual and G-isotropic codes are invariant under the extended Clifford-Weil groups. A reader cares because the results unify classical coding invariants with symmetry constraints in a broad algebraic framework.

Core claim

By defining transformation groups for G-full weight enumerators that extend the Clifford-Weil groups, the paper proves that the G-full weight enumerators of G-self-dual and G-isotropic codes are invariant under these groups and span the invariant subspaces of these groups.

What carries the argument

The G-full weight enumerator under the action of the extended Clifford-Weil group generated by G-quadratic maps and G-representations.

If this is right

G-analogues of the MacWilliams identity hold for the G-full weight enumerators of G-codes.
The G-full weight enumerators of G-self-dual and G-isotropic codes generate the full space of invariants under the extended Clifford-Weil groups.
The framework applies uniformly to any finite ring and any bilinear form without further restrictions.
G-self-dual codes produce all invariants in the G-setting.

Where Pith is reading between the lines

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

The results supply an algorithmic route to enumerate all G-invariant self-dual codes by computing bases of Clifford-Weil invariants.
Specific choices of G, such as cyclic or dihedral groups, could yield new families of symmetric codes over rings like Z/4Z.
The same G-representation machinery may transfer to the study of group-invariant lattices or designs.

Load-bearing premise

The generalizations of Hayden's theorem, Atsumi's MacWilliams identity, and Gleason-type theorems continue to hold when the underlying ring, bilinear form, and permutation group G are taken arbitrarily.

What would settle it

A concrete finite ring, bilinear form, and permutation group G for which the G-full weight enumerator of some G-self-dual code fails to be invariant under the corresponding Clifford-Weil group.

read the original abstract

In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the concept of $G$-dual codes for codes invariant under a permutation group $G$, referred to as $G$-codes. We then present several generalizations of Atsumi's MacWilliams identity (Atsumi, 1995; Chakraborty and Miezaki, 2023) for $G$-codes over finite rings with respect to general bilinear forms. Furthermore, we establish a $G$-analogue of the MacWilliams identity for $G$-full weight enumerators and introduce the notions of $G$-quadratic maps and $G$-representations for twisted modules, twisted rings, quadratic pairs, and form rings. By defining transformation groups for $G$-full weight enumerators, we extend the theory of Clifford--Weil groups (Nebe, Rains, and Sloane, 2004, 2006). Finally, we provide generalizations of Gleason-type theorems for these weight enumerators, demonstrating that the $G$-full weight enumerators of $G$-self-dual and $G$-isotropic codes are invariant under the Clifford--Weil groups and span the invariant subspaces of these groups.

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

Extends MacWilliams and Gleason theorems to G-invariant codes over arbitrary rings, but the ring generality looks like the part that needs checking.

read the letter

The main point here is a systematic extension of the classical results on self-dual codes to the G-invariant setting. The paper defines G-dual codes with respect to general bilinear forms, introduces G-quadratic maps and G-representations, and gives a G-version of the MacWilliams identity for full weight enumerators. It then claims that these enumerators are invariant under generalized Clifford-Weil groups and that they span the invariant subspaces for G-self-dual and G-isotropic codes. That is the concrete new content, built on top of Hayden, Atsumi, and the Nebe-Rains-Sloane framework.

Referee Report

2 major / 2 minor

Summary. The paper defines dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and generalizes Hayden's theorem. It introduces G-dual codes for G-invariant codes (G-codes), provides several generalizations of Atsumi's MacWilliams identity for G-codes, defines a G-analogue for G-full weight enumerators along with G-quadratic maps and G-representations for twisted modules/rings/quadratic pairs/form rings, extends the theory of Clifford-Weil groups via transformation groups for the enumerators, and generalizes Gleason-type theorems to conclude that the G-full weight enumerators of G-self-dual and G-isotropic codes are invariant under the Clifford-Weil groups and span the invariant subspaces.

Significance. If the central generalizations hold, the work would provide a unified extension of classical coding-theoretic results (Hayden, Atsumi, Nebe-Rains-Sloane) to include arbitrary permutation group actions and general bilinear forms over arbitrary finite rings. The spanning property for the invariant subspaces would be a useful tool for classification via invariant theory, and the new notions of G-quadratic maps and twisted representations could enable further constructions beyond standard fields or Frobenius rings.

major comments (2)

[§4] §4 (generalizations of Atsumi's MacWilliams identity and the G-analogue): The claim of validity for arbitrary finite rings (without restriction to Frobenius rings) is load-bearing for all subsequent results on G-full weight enumerators and the spanning statement. Standard character-sum or module-isomorphism proofs of MacWilliams identities require the ring to be Frobenius so that |C^⊥| = |C| and the annihilator behaves appropriately; this fails for rings such as ℤ/4ℤ × ℤ/2ℤ. The manuscript must either add an explicit restriction on the rings or supply a proof that avoids the Frobenius hypothesis.
[§5] §5 (Gleason-type theorems and spanning claim): The assertion that the G-full weight enumerators span the invariant subspaces of the (generalized) Clifford-Weil groups rests on the preceding MacWilliams and invariance results. If the ring generality in §4 does not hold, this spanning statement cannot be established for arbitrary inputs.

minor comments (2)

[Introduction] The abstract introduces a large number of new terms (G-dual codes, G-quadratic maps, G-representations, twisted modules, etc.) in rapid succession; a short table or diagram in the introduction summarizing the relationships among these objects would improve readability.
Notation for the bilinear forms and the action of G is introduced without an explicit summary of all standing assumptions; a dedicated 'Notation and assumptions' paragraph early in the paper would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and detailed report. The comments highlight an important technical point regarding the scope of our results, and we address each major comment below.

read point-by-point responses

Referee: [§4] §4 (generalizations of Atsumi's MacWilliams identity and the G-analogue): The claim of validity for arbitrary finite rings (without restriction to Frobenius rings) is load-bearing for all subsequent results on G-full weight enumerators and the spanning statement. Standard character-sum or module-isomorphism proofs of MacWilliams identities require the ring to be Frobenius so that |C^⊥| = |C| and the annihilator behaves appropriately; this fails for rings such as ℤ/4ℤ × ℤ/2ℤ. The manuscript must either add an explicit restriction on the rings or supply a proof that avoids the Frobenius hypothesis.

Authors: We agree with the referee that the standard proofs of MacWilliams-type identities rely on the ring being Frobenius. Our generalization of Atsumi's identity for G-codes over arbitrary finite rings with respect to general bilinear forms uses character-sum techniques that implicitly require |C^⊥| = |C| and bijective annihilator maps, which do not hold for non-Frobenius rings such as ℤ/4ℤ × ℤ/2ℤ. We do not possess a proof that removes the Frobenius hypothesis. In the revised manuscript we will therefore add an explicit restriction to Frobenius rings in the statements of all theorems in §4 (and update the abstract and introduction accordingly), while preserving the new G-analogue constructions and G-quadratic maps for this class of rings. revision: yes
Referee: [§5] §5 (Gleason-type theorems and spanning claim): The assertion that the G-full weight enumerators span the invariant subspaces of the (generalized) Clifford-Weil groups rests on the preceding MacWilliams and invariance results. If the ring generality in §4 does not hold, this spanning statement cannot be established for arbitrary inputs.

Authors: The referee is correct that the Gleason-type theorems and the spanning property for the invariant subspaces in §5 are direct consequences of the MacWilliams identities and invariance results established in §4. Once the restriction to Frobenius rings is incorporated in §4, the statements in §5 will be revised to hold under the same hypothesis. We will also insert a brief remark at the beginning of §5 making the logical dependence explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: generalizations of external theorems with independent definitions

full rationale

The paper defines dual codes and G-codes over arbitrary finite rings with respect to arbitrary bilinear forms, then claims to prove generalizations of Hayden's theorem, Atsumi's MacWilliams identity, and Gleason-type theorems for G-full weight enumerators and Clifford-Weil groups. These steps are presented as new derivations extending cited prior results (Bridges-Hall-Hayden 1981, Atsumi 1995, Nebe-Rains-Sloane 2004/2006, Chakraborty-Miezaki 2023) rather than reductions of outputs to the paper's own fitted quantities or self-citations. No self-definitional loops, fitted-input predictions, or load-bearing self-citation chains appear in the abstract or described chain; the central spanning claim follows from the stated generalizations, which are treated as independently established. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard algebraic assumptions about finite rings, bilinear forms, and permutation group actions; no numerical free parameters appear. New concepts are introduced by definition rather than as postulated entities with independent evidence.

axioms (2)

domain assumption Finite rings admit arbitrary bilinear forms that define duality.
Invoked in the opening definition of dual codes over arbitrary finite rings.
domain assumption Permutation groups act on code coordinates while preserving the code.
Central to the definition of G-codes and G-dual codes.

pith-pipeline@v0.9.0 · 5535 in / 1315 out tokens · 44317 ms · 2026-05-08T18:05:53.567241+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/Cost/FunctionalEquation.lean (J = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

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

G-Hamming weight enumerator hwe_G(θC)(x,y) = (1/|θ(C^⊥M)|) hwe_G(θ(C^⊥M))(x+(|V|-1)y, x-y).
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection (coupling combiner forces bilinear branch) unclear

?

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

Clifford–Weil group C(ρ_G) generated by m_r, d_φ, h_{ι,μ,ν}; G-full weight enumerators of G-self-dual G-isotropic codes are invariant under C(ρ_G).
IndisputableMonolith/Foundation/AlexanderDuality.lean (D=3 forcing) — unrelated structural duality alexander_duality_circle_linking unclear

?

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

θ = (1/|G|) Σ_{g∈G} g idempotent; (θC)^⊥M = ker θ^T ⊕ θ^T (C^⊥M).

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

16 extracted references · 10 canonical work pages

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