When isometry and equivalence for skew constacyclic codes coincide

Monica Nevins; Susanne Pumpluen

arxiv: 2508.06695 · v3 · submitted 2025-08-08 · 💻 cs.IT · math.IT· math.RA

When isometry and equivalence for skew constacyclic codes coincide

Monica Nevins , Susanne Pumpluen This is my paper

Pith reviewed 2026-05-18 23:39 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.RA

keywords skew constacyclic codesisometryequivalencePetit ringsnonassociative ringsHamming weightcode classification

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The pith

For most skew constacyclic codes, isometry and equivalence coincide because weight-preserving isomorphisms of their ambient rings have degree one.

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

The paper shows that notions of isometry and equivalence coincide for most skew constacyclic codes over a commutative base ring. It establishes this by proving that Hamming-weight preserving isomorphisms between the ambient rings must have degree one when the rings are non-associative. The ambient rings are the nonassociative Petit rings obtained as quotients of skew polynomial rings. The result leads to new definitions of equivalence and isometry that capture all such isomorphisms and produce tighter classifications of the codes.

Core claim

The notions of (n,σ)-isometry and (n,σ)-equivalence coincide for most skew (σ,a)-constacyclic codes of length n over a commutative base ring S. This follows from showing that all Hamming-weight preserving isomorphisms between their ambient rings which extend some automorphism τ of S that commutes with σ must have degree one when those rings are not associative. In the process the isomorphisms between their nonassociative ambient rings, the Petit rings S[t;σ]/(t^n-a), are determined. As a consequence new definitions of equivalence and isometry are proposed that exactly capture all Hamming-weight preserving isomorphisms between the ambient rings which extend τ in Aut(S) that commute with σ and

What carries the argument

Hamming-weight preserving isomorphisms of degree one between nonassociative Petit rings that extend a commuting automorphism of the base ring.

If this is right

Isometry and equivalence are identical for most of these codes.
New definitions of equivalence and isometry capture exactly the weight-preserving isomorphisms.
The new definitions produce tighter classifications of skew constacyclic codes.
Isomorphisms between the nonassociative ambient rings are completely determined.

Where Pith is reading between the lines

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

Classifications of skew constacyclic codes can treat isometry and equivalence as one concept in the covered cases.
The degree-one restriction may distinguish nonassociative rings from associative ones in similar code settings.
The method of determining ring isomorphisms could apply to other families of linear codes with nonassociative ambient structures.

Load-bearing premise

All Hamming-weight preserving isomorphisms between the ambient rings extend some automorphism τ of S that commutes with σ, and the rings are non-associative.

What would settle it

A Hamming-weight preserving isomorphism between two such ambient rings that extends a commuting automorphism of S but has degree greater than one would show the notions do not coincide.

read the original abstract

We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. We show that the notions of $(n,\sigma)$-isometry and $(n,\sigma)$-equivalence introduced by Ou-azzou et al coincide for most skew $(\sigma,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight preserving isomorphisms between their ambient rings which extend some automorphism $\tau$ of $S$ that commutes with $\sigma$ must have degree one, when those rings are not associative. In the process we determine isomorphisms between their nonassociative ambient rings, the Petit rings $S[t;\sigma]/S[t;\sigma](t^n-a)$, which give rise to skew constacyclic codes. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-weight preserving isomorphisms between the ambient rings of skew constacyclic codes which extend $\tau\in {\rm Aut}(S)$ that commute with $\sigma$, and lead to tighter classifications.

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 shows that weight-preserving isomorphisms extending a commuting automorphism must have degree one in the non-associative case, which makes isometry and equivalence coincide under that condition and yields tighter definitions.

read the letter

The main point is that this paper proves the two notions coincide for most skew constacyclic codes when the ambient rings are non-associative. They establish that any Hamming-weight preserving isomorphism between the Petit rings that extends a suitable commuting automorphism τ of the base ring must have degree one. This directly addresses the definitions introduced by Ou-azzou et al. and gives a cleaner algebraic handle on the maps that preserve the code structure.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that (n,σ)-isometry and (n,σ)-equivalence coincide for most skew (σ,a)-constacyclic codes of length n over a commutative ring S. The central argument shows that any Hamming-weight-preserving ring isomorphism between the ambient non-associative Petit rings S[t;σ]/(t^n−a) that extends an automorphism τ of S commuting with σ must have degree one; new definitions of equivalence and isometry are then proposed to capture precisely those isomorphisms.

Significance. If the central claim holds, the work tightens the classification of skew constacyclic codes by equating two previously distinct notions of equivalence, which should simplify the study of their automorphism groups and minimum-distance properties. The explicit determination of isomorphisms between non-associative ambient rings is a concrete technical contribution that may be reusable in related settings.

major comments (1)

[Main proof / abstract statement] The main proof (abstract and §3) establishes the degree-one property only under the additional hypothesis that the isomorphism already extends a commuting τ ∈ Aut(S). It does not derive the extension property itself from Hamming-weight preservation. Because the definition of (n,σ)-isometry uses arbitrary weight-preserving isomorphisms while (n,σ)-equivalence uses only those induced by degree-one maps extending τ, this gap is load-bearing for the claimed coincidence.

minor comments (2)

[Introduction / §2] The precise conditions on S and σ that make the Petit ring non-associative should be stated explicitly with a short example or reference to a concrete parameter set.
[Preliminaries] Notation for the quotient ring S[t;σ]/(t^n−a) is introduced in several places; a single consolidated definition early in the paper would improve readability.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading and for identifying this important distinction in our argument. We address the major comment below and will make a partial revision to clarify the scope of the result and the role of the proposed new definitions.

read point-by-point responses

Referee: [Main proof / abstract statement] The main proof (abstract and §3) establishes the degree-one property only under the additional hypothesis that the isomorphism already extends a commuting τ ∈ Aut(S). It does not derive the extension property itself from Hamming-weight preservation. Because the definition of (n,σ)-isometry uses arbitrary weight-preserving isomorphisms while (n,σ)-equivalence uses only those induced by degree-one maps extending τ, this gap is load-bearing for the claimed coincidence.

Authors: We agree that the proof in §3 assumes the isomorphism extends a commuting automorphism τ of S and shows that any such map must have degree one when the Petit ring is non-associative. We do not establish that Hamming-weight preservation alone forces the existence of an extending τ. This means the original notions of (n,σ)-isometry and (n,σ)-equivalence (as defined by Ou-azzou et al.) are not shown to coincide for completely arbitrary weight-preserving isomorphisms. However, the manuscript already proposes new definitions of equivalence and isometry that are defined exactly as the Hamming-weight-preserving isomorphisms extending a commuting τ. Under these revised definitions the two notions coincide by the degree-one theorem. We will revise the abstract, introduction, and the statement of the main result to make explicit that the coincidence holds for the class of maps extending τ and that the new definitions are introduced precisely to capture this class. This constitutes a partial revision focused on clarification rather than a full closure of the gap for arbitrary isomorphisms. revision: partial

standing simulated objections not resolved

Deriving from Hamming-weight preservation alone that every isomorphism between the ambient non-associative Petit rings extends some automorphism τ of S that commutes with σ.

Circularity Check

0 steps flagged

Direct algebraic proof with no reduction to self-referential inputs or fitted quantities

full rationale

The manuscript establishes that Hamming-weight-preserving isomorphisms extending a commuting automorphism τ must have degree one when the Petit ring is non-associative, then refines the definitions of isometry and equivalence to match precisely those maps. This is a self-contained algebraic argument over skew polynomial rings with no parameter fitting, no self-citation chains that bear the central load, and no redefinition of inputs as outputs. The derivation does not collapse by construction; the extension property is an explicit hypothesis rather than a derived claim that loops back to the result itself. The paper is therefore free of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a pure proof inside the established theory of skew polynomial rings and non-associative quotients; no numerical parameters are fitted and no new entities are postulated.

axioms (1)

standard math Standard algebraic properties of skew polynomial rings, Petit quotients, and automorphisms of the base ring S
The proof relies on known facts about these structures rather than new axioms introduced in the paper.

pith-pipeline@v0.9.0 · 5723 in / 1250 out tokens · 43247 ms · 2026-05-18T23:39:07.871441+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/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Theorem (Corollary 4.5). ... all nonzero Hamming weight preserving homomorphisms G between them such that G|K commutes with σ are monomial of degree one ... when those algebras are proper nonassociative.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Theorem 3.3. The monomial αt^k ... is power-associative if and only if αb=σ^n(α)σ^k(b).

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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