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arxiv: 2504.09098 · v2 · submitted 2025-04-12 · 💻 cs.IT · math.IT· math.RA

The trace dual of nonlinear skew cyclic codes

Daniel Bossaller , Daniel Herden , Indalecio Ruiz-Bola\~nos This is my paper

Pith reviewed 2026-05-22 21:08 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.RA
keywords trace dualskew cyclic codescyclic codesfinite fieldstrace Euclidean dualtrace Hermitian dualF_q-linear codesskew polynomial ring
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The pith

For odd prime powers q, the trace Euclidean and trace Hermitian duals of F_q-linear cyclic and skew cyclic codes over F_{q^2} are determined by explicit generators.

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

This paper works out explicit generators for the trace duals of codes that are linear only over a subfield F_q but live in the quadratic extension F_{q^2}. A sympathetic reader would care because these duals appear in constructions of quantum error-correcting codes. The central focus is on ordinary cyclic codes that are F_q-linear over F_{q^2} and on the skew-cyclic variant where the ring multiplication is twisted by the Frobenius map. If the formulas hold, they give a direct way to write down the dual code from the original code's generator without enumerating the whole dual space.

Core claim

Given the field extension F_q ≤ F_{q^2} with q an odd prime power, the trace Euclidean and trace Hermitian dual codes for the general F_q-linear cyclic F_{q^2}-code are determined; the same is done for the trace Euclidean and trace Hermitian duals of general F_q-linear skew cyclic F_{q^2}-codes, defined as left F_q[X]-submodules of F_{q^2}[X;σ]/(X^n-1) where σ is the Frobenius automorphism.

What carries the argument

The left F_q[X]-submodule structure inside the quotient of the skew polynomial ring F_{q^2}[X;σ]/(X^n-1), which carries the argument by letting the dual generators be read off from the original module generators once q is odd.

If this is right

  • The trace dual of any such cyclic code is itself generated by an explicitly constructible element in the same module.
  • Both the Euclidean and Hermitian trace duals admit similar closed-form descriptions in terms of the original generator.
  • The same generator-finding procedure applies uniformly to the skew-cyclic family.
  • The oddness of q is required for the trace maps to interact cleanly with the module structure.

Where Pith is reading between the lines

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

  • Having explicit dual generators could make it easier to locate self-orthogonal codes inside these families for quantum-code applications.
  • The method might adapt to higher-degree extensions if an analogous module description is available.
  • Direct verification on small fields and short lengths would quickly confirm or refute the generator formulas.

Load-bearing premise

That the module definition over the skew polynomial ring together with q being odd is enough to produce explicit generators for the dual codes.

What would settle it

Pick small values such as q=3 and n=5, generate a specific F_3-linear skew cyclic code over F_9, compute its trace Euclidean dual by direct linear algebra, and check whether the paper's stated generator produces exactly that dual space.

read the original abstract

Codes which have a finite field $\mathbb{F}_{q^m}$ as their alphabet but which are only linear over a subfield $\mathbb{F}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^2}$. In particular, given the field extension $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general $\mathbb{F}_q$-linear cyclic $\mathbb{F}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general $\mathbb{F}_q$-linear skew cyclic $\mathbb{F}_{q^2}$-codes, which are defined to be left $\mathbb{F}_q[X]$-submodules of $\mathbb{F}_{q^2}[X;\sigma]/(X^n-1)$, where $\sigma$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;\sigma]$ the induced skew polynomial ring.

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

2 major / 2 minor

Summary. The manuscript determines explicit generators for the trace Euclidean and trace Hermitian duals of general F_q-linear cyclic F_{q^2}-codes and of general F_q-linear skew cyclic F_{q^2}-codes (defined as left F_q[X]-submodules of F_{q^2}[X;σ]/(X^n-1) with σ the Frobenius automorphism), for odd prime power q.

Significance. Explicit dual generators for this class of subfield-linear codes would be useful for quantum code constructions and for understanding duality in skew polynomial modules. The module-theoretic framing is a natural extension of prior work on cyclic codes over finite fields.

major comments (2)
  1. [Derivation of duals for skew cyclic codes (general case)] The central claim that dual generators can be read off directly from a generator of C assumes the trace form remains non-degenerate on the quotient and that the adjoint induced by the trace preserves the left F_q[X]-module structure. The manuscript does not visibly address the case gcd(n, q-1) > 1, where the Frobenius action on roots may fail to commute appropriately with the trace map; if this occurs, the stated generators would not span the dual.
  2. [Trace Hermitian dual construction] The oddness of q is used to guarantee that the Hermitian form is well-defined, but the argument does not check whether the resulting dual remains a left F_q[X]-submodule when the minimal polynomial of the generator does not split in a manner compatible with the adjoint operation.
minor comments (2)
  1. An explicit small example (e.g., n=3 or n=5 with a concrete generator polynomial) would help verify that the dual generator formula produces a code whose trace dual matches the claimed span.
  2. Notation for the skew polynomial ring and the precise definition of the trace map should be recalled in the section introducing the duals to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying points that require clarification. We address each major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [Derivation of duals for skew cyclic codes (general case)] The central claim that dual generators can be read off directly from a generator of C assumes the trace form remains non-degenerate on the quotient and that the adjoint induced by the trace preserves the left F_q[X]-module structure. The manuscript does not visibly address the case gcd(n, q-1) > 1, where the Frobenius action on roots may fail to commute appropriately with the trace map; if this occurs, the stated generators would not span the dual.

    Authors: We agree that the non-degeneracy and module-preservation arguments merit explicit verification in the general case. The trace form is non-degenerate on the quotient ring for any n because the trace from F_{q^2} to F_q is surjective and the Frobenius automorphism is bijective; the adjoint operation is defined via the skew multiplication in F_{q^2}[X;σ] and therefore automatically preserves the left F_q[X]-module structure. Nevertheless, the manuscript does not contain a dedicated paragraph treating the subcase gcd(n,q-1)>1. We will add a short remark (or lemma) confirming that the same generator formulas remain valid when the cyclotomic cosets are not of full size, thereby addressing the referee’s concern directly. revision: yes

  2. Referee: [Trace Hermitian dual construction] The oddness of q is used to guarantee that the Hermitian form is well-defined, but the argument does not check whether the resulting dual remains a left F_q[X]-submodule when the minimal polynomial of the generator does not split in a manner compatible with the adjoint operation.

    Authors: The oddness of q ensures that the Hermitian inner product is non-degenerate and that the adjoint map is well-defined on the skew polynomial ring. Because the dual is constructed by applying the adjoint to the generator polynomial and then taking the appropriate left ideal in the quotient, the resulting code is automatically a left F_q[X]-submodule by the ring homomorphism properties of the adjoint. We acknowledge that the manuscript does not spell out the compatibility when the minimal polynomial is irreducible over F_{q^2}; we will insert a brief verification that the adjoint operation commutes with left multiplication by elements of F_q[X] regardless of splitting behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses direct module-theoretic arguments

full rationale

The paper determines generators for trace Euclidean and trace Hermitian duals of F_q-linear cyclic and skew-cyclic F_{q^2}-codes by working directly with left F_q[X]-submodules of the quotient ring F_{q^2}[X;σ]/(X^n-1) under the Frobenius automorphism σ. The abstract and provided excerpts contain no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central claims to prior unverified results by the same authors. The oddness of q is used only to ensure the Hermitian form is well-defined; the explicit dual generators are obtained from the module structure without circular reduction to the inputs. The derivation is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard finite-field and skew-polynomial axioms; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math F_q is a subfield of F_{q^2} equipped with the Frobenius automorphism σ
    Standard property of finite-field extensions invoked to define the skew multiplication.
  • standard math The quotient ring F_{q^2}[X;σ]/(X^n-1) carries a natural left F_q[X]-module structure
    Definition of skew-cyclic codes as submodules.

pith-pipeline@v0.9.0 · 5750 in / 1257 out tokens · 46916 ms · 2026-05-22T21:08:42.093213+00:00 · methodology

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