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arxiv: 2606.02137 · v1 · pith:77AL7ZZPnew · submitted 2026-06-01 · 💻 cs.IT · math.IT

QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over mathbb{F}₂ times (mathbb{F}₂+vmathbb{F}₂)

Rabia Zengin , Mehmet Emin K\"oro\u{g}lu This is my paper

Pith reviewed 2026-06-28 12:59 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords cyclic codesHermitian hullsHermitian sumsquantum error-correcting codesentanglement-assisted quantum codesmatrix product codeslinear complementary dual codescomposite rings
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The pith

Cyclic codes over F2 × (F2 + vF2) yield QEC codes via their Hermitian hulls and sums using the Hermitian dual of Quantum Construction X.

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

The paper computes explicit generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes over the ring F2 × (F2 + vF2) with v squared equal to v. These structures are then fed into the Hermitian dual version of Quantum Construction X to produce quantum error-correcting codes. Separately, linear complementary dual codes over the same ring are combined via matrix product constructions to obtain families of entanglement-assisted quantum error-correcting codes.

Core claim

By determining the generator polynomials of Hermitian hulls and sums of cyclic codes over F2 × (F2 + vF2), the Hermitian dual version of Quantum Construction X can be applied directly to these objects to obtain QEC codes, while matrix product methods on LCD codes over the ring produce EAQEC codes.

What carries the argument

Hermitian hulls and sums of cyclic codes over the ring, to which the Hermitian dual version of Quantum Construction X is applied.

If this is right

  • Explicit generator polynomials for Hermitian hulls and sums become available for all cyclic codes over this ring.
  • New QEC codes arise whenever the hull or sum satisfies the dimension conditions for Quantum Construction X.
  • EAQEC codes are obtained whenever an LCD code over the ring is combined with a matrix product construction.
  • The resulting code parameters are determined by the classical parameters of the hull, sum, or LCD code.

Where Pith is reading between the lines

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

  • The same ring constructions might be applied to non-cyclic codes or to other composite rings to enlarge the pool of obtainable QEC and EAQEC parameters.
  • The explicit generator polynomials could be used to search for codes with improved distance properties by varying the underlying cyclic code length.
  • If the ring admits a natural embedding into larger alphabets, the codes might lift to quantum codes over higher-dimensional systems.

Load-bearing premise

That the Hermitian dual version of Quantum Construction X applies directly to the computed Hermitian hulls and sums over this ring and produces valid QEC codes with the expected parameters.

What would settle it

An explicit cyclic code over the ring whose computed Hermitian hull, when fed into the stated construction, produces a quantum code whose minimum distance or dimension fails to match the formula predicted by the hull parameters.

read the original abstract

In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$, where $v^2 = v$. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.

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 manuscript determines the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes over the ring R = F₂ × (F₂ + v F₂) with v² = v. It then applies the Hermitian dual version of Quantum Construction X to these structures to obtain QEC codes and employs matrix product code methods on LCD codes over the same ring to derive families of EAQEC codes.

Significance. If the constructions are valid and the resulting codes satisfy the required quantum conditions, the work would supply explicit algebraic generators for new QEC and EAQEC families derived from cyclic codes over a composite ring, extending quantum coding constructions beyond fields and potentially yielding codes with computable parameters from the classical hull dimensions.

major comments (1)
  1. [Abstract (workflow description)] The central claim that the Hermitian dual version of Quantum Construction X applied to the computed Hermitian hulls and sums over R produces valid QEC codes (with parameters following from the classical data) is load-bearing, yet the manuscript supplies no explicit verification that the output stabilizer generators commute under the Hermitian inner product or that the quantum distance meets the expected lower bound; this is especially pertinent because R is not a field and standard statements of Construction X assume a field base.
minor comments (1)
  1. The abstract would be strengthened by the inclusion of at least one concrete example with explicit generator polynomials, code parameters, and distance values to illustrate the constructions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for stronger justification of the central construction. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract (workflow description)] The central claim that the Hermitian dual version of Quantum Construction X applied to the computed Hermitian hulls and sums over R produces valid QEC codes (with parameters following from the classical data) is load-bearing, yet the manuscript supplies no explicit verification that the output stabilizer generators commute under the Hermitian inner product or that the quantum distance meets the expected lower bound; this is especially pertinent because R is not a field and standard statements of Construction X assume a field base.

    Authors: We agree that the manuscript relies on the extension of the Hermitian dual version of Quantum Construction X to the ring R without supplying explicit verification of commutation or distance bounds in the text. Because R is a finite commutative Frobenius ring (isomorphic to a product of fields via the Chinese Remainder Theorem), the Hermitian inner product is well-defined componentwise, and the standard commutation and distance arguments carry over directly from the field components. Nevertheless, to meet the referee's concern we will add a dedicated subsection in the revision that (i) states the precise extension of Construction X to this ring, (ii) proves that Hermitian orthogonality of the hull/sum codes implies the required stabilizer commutation, and (iii) includes explicit parameter checks for representative codes confirming the quantum distance lower bound. This revision will be made. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper computes generator polynomials for Hermitian hulls and sums of cyclic codes over R = F2 × (F2 + v F2), then applies the Hermitian dual version of Quantum Construction X and matrix product methods on LCD codes. These steps use standard external constructions on explicitly derived inputs without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citation chains. No equations or claims reduce the output parameters to the inputs by construction; the central results are independent of the paper's own fitted values or prior self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger left empty pending full text.

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Forward citations

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