Pith. sign in

REVIEW 2 major objections 2 minor 33 references

Hulls and sums of separable constacyclic codes over the ring S = F_q × (F_q + v F_q) produce new quantum error-correcting codes with improved parameters.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-26 11:11 UTC pith:ON43PSVJ

load-bearing objection The paper gives explicit generator polynomials for the duals, hulls, and sums of separable constacyclic codes over this product ring together with their Gray images, then uses those to claim new quantum codes with improved parameters. the 2 major comments →

arxiv 2606.22069 v1 pith:ON43PSVJ submitted 2026-06-20 cs.IT math.IT

Hulls and sums of separable constacyclic codes over mathbb{F}_q times (mathbb{F}_q+vmathbb{F}_q) and new quantum codes

Yu Qian , Yu Wang , Liqi Wang This is my paper
classification cs.IT math.IT
keywords constacyclic codeshullssumsGray imagesquantum error-correcting codesseparable codesproduct rings
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper derives explicit generator polynomials for the Euclidean and Hermitian duals, hulls, and sums of separable constacyclic codes over the ring S, together with the generator polynomials of their Gray images. It then presents two construction methods that turn these hulls and sums into quantum error-correcting codes. The resulting codes are stated to have better parameters than previously known constructions. A reader would care because improved quantum codes directly affect the distance and dimension achievable in quantum error correction.

Core claim

The generator polynomials of the Euclidean and Hermitian hulls and sums of separable constacyclic codes over S and of their Gray images are given explicitly; two methods are proposed that convert these objects into quantum error-correcting codes whose parameters exceed those of existing tables.

What carries the argument

The generator polynomials of the Euclidean and Hermitian hulls and sums (and their Gray images), which determine the quantum code parameters via the two proposed construction methods.

Load-bearing premise

The Gray images of the hulls and sums satisfy the length and dimension conditions needed for the CSS or Hermitian quantum constructions, and the claimed parameter improvements are accurate against existing bounds.

What would settle it

A concrete separable constacyclic code over S whose Gray-image hull or sum produces a quantum code whose minimum distance or dimension falls short of the values listed in the paper's tables or violates a known quantum bound.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The two methods convert any separable constacyclic code over S whose hull or sum meets the required conditions into a quantum code.
  • The explicit generator polynomials allow direct computation of the parameters of the resulting quantum codes.
  • New quantum codes are obtained whose parameters improve on those recorded in prior tables for the same length and dimension.

Where Pith is reading between the lines

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

  • The same hull-and-sum approach could be tested on non-separable constacyclic codes over the same ring to see whether the improvement persists.
  • If the Gray-image step preserves the hull structure for other rings of the form F_q × R, the methods might extend beyond the specific ring S studied here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript derives generator polynomials for the Euclidean and Hermitian duals of separable constacyclic codes over the ring S = F_q × (F_q + v F_q) (v² = v, q odd prime power), along with the generator polynomials of their Gray images. It then obtains the generator polynomials for the Euclidean and Hermitian hulls and sums of these codes and their Gray images. Finally, it proposes two methods to construct quantum error-correcting codes from the hulls and sums, claiming to produce new QECCs with improved parameters over existing constructions.

Significance. If the derivations are correct and the Gray images of the hulls/sums satisfy the self-orthogonality conditions needed for the CSS and Hermitian quantum constructions, the work supplies explicit algebraic constructions for new families of quantum codes. Explicit generator polynomials and the focus on separable constacyclic codes over this ring extension are standard tools that can yield reproducible families; credit is due for attempting to produce parameter improvements via hulls and sums.

major comments (2)
  1. [final section on quantum constructions] The quantum-code claims rest on the Gray images satisfying C ⊆ C^⊥ (or the Hermitian analogue) with the stated dimension and distance. The manuscript must explicitly verify that the coordinate-wise Gray map from S to F_q² preserves the relevant Euclidean/Hermitian inner-product relations for the constacyclic multipliers appearing in the generator polynomials; without this step the CSS/Hermitian constructions do not apply and the listed parameter improvements cannot be claimed (see the construction of the two methods in the final section).
  2. [final section on quantum constructions] The assertion that the new QECCs 'outperform the existing ones in terms of parameters' requires concrete tables or explicit comparisons against known bounds or tables of best-known quantum codes; the abstract supplies no such verification, and any such tables must be checked for correctness of the distance and dimension calculations.
minor comments (2)
  1. All generator polynomials should be accompanied by short derivation sketches or references to the standard results on constacyclic codes over product rings that are being extended.
  2. Notation for the ring S and the Gray map should be introduced once and used consistently; ensure that the definition of the Gray map appears before its first use in the dual/hull statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [final section on quantum constructions] The quantum-code claims rest on the Gray images satisfying C ⊆ C^⊥ (or the Hermitian analogue) with the stated dimension and distance. The manuscript must explicitly verify that the coordinate-wise Gray map from S to F_q² preserves the relevant Euclidean/Hermitian inner-product relations for the constacyclic multipliers appearing in the generator polynomials; without this step the CSS/Hermitian constructions do not apply and the listed parameter improvements cannot be claimed (see the construction of the two methods in the final section).

    Authors: We agree that an explicit verification is required. The manuscript derives the relevant generator polynomials but does not contain a dedicated proof that the Gray map preserves the Euclidean and Hermitian inner products for the constacyclic multipliers in question. In the revised manuscript we will insert a lemma establishing this preservation property, thereby justifying the application of the CSS and Hermitian constructions to the Gray images of the hulls and sums. revision: yes

  2. Referee: [final section on quantum constructions] The assertion that the new QECCs 'outperform the existing ones in terms of parameters' requires concrete tables or explicit comparisons against known bounds or tables of best-known quantum codes; the abstract supplies no such verification, and any such tables must be checked for correctness of the distance and dimension calculations.

    Authors: We accept that concrete parameter comparisons are necessary to substantiate the improvement claim. The revised manuscript will include tables that list the parameters of the new quantum codes, compare them with existing constructions from the literature, and reference known quantum-code bounds; all distance and dimension calculations will be double-checked before inclusion. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations use standard dual/hull/sum formulas and external quantum constructions

full rationale

The abstract and available text describe computing generator polynomials for Euclidean/Hermitian duals, hulls and sums of separable constacyclic codes over the given ring, followed by their Gray images, then applying standard CSS/Hermitian constructions to produce QECCs. No quoted equations reduce a claimed prediction or result to a fitted input or self-citation by construction. No self-citations appear in the provided text, and the central claims rest on algebraic derivations that are independent of the target QECC parameters. This is the normal case of a self-contained paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the constructions appear to rest on standard algebraic properties of finite rings and Gray maps already present in the literature.

pith-pipeline@v0.9.1-grok · 5701 in / 1118 out tokens · 28388 ms · 2026-06-26T11:11:29.948825+00:00 · methodology

0 comments
read the original abstract

We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over $\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$, with $q$ an odd prime power and $v^2=v$, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over $\mathcal{S}$ and their Gray images are presented, respectively. Furthermore, we provide the generator polynomials of the Euclidean sums and Hermitian sums of separable constacyclic codes and their Gray images, respectively. Finally, we propose two methods to yield quantum error-correcting codes (QECCs) from the hulls and sums of separable constacyclic codes over $\mathcal{S}$, and generate new QECCs that outperform the existing ones in terms of parameters.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references

  1. [1]

    R., Rains, E

    Calderbank, A. R., Rains, E. M., Shor, P. M., Sloane, N.: Quantum error correction via codes over GF(4). IEEE Transactions on Information Theory . 44(4), 1369-1387 (1998)

    1998

  2. [2]

    IEEE Transactions on Infor- mation Theory

    Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Transactions on Infor- mation Theory. 47(7), 3065-3072 (2001)

    2001

  3. [3]

    Designs, Codes and Cryptography

    Lison ˇek, P., Singh, V .: Quantum codes from nearly self-orthogonal quaternary linear codes. Designs, Codes and Cryptography. 73(2), 417-424 (2014)

    2014

  4. [4]

    Quantum Information Processing

    Gao, J., Wang, Y .:u-Constacyclic codes overF p +uF p and their applications of constructing new non-binary quantum codes. Quantum Information Processing. 17(1), 4 (2018) 21

    2018

  5. [5]

    W.: Constacyclic codes over the ringF q +vF q +v 2Fq and their appli- cations of constructing new non-binary quantum codes

    Ma, F., Gao, J., Fu, F. W.: Constacyclic codes over the ringF q +vF q +v 2Fq and their appli- cations of constructing new non-binary quantum codes. Quantum Information Processing. 17(6), 122 (2018)

    2018

  6. [6]

    Applicable Algebra in Engineering, Communication and Computing

    Gao, Y ., Gao, J., Fu, F.: Quantum codes from cyclic codes over the ringF q +v 1Fq +· · ·+ vrFq. Applicable Algebra in Engineering, Communication and Computing. 30(2), 161-174 (2019)

    2019

  7. [7]

    Quantum Information Processing

    Wang, Y ., Kai, X., Sun, Z., Zhu, S.: Quantum codes from Hermitian dual-containing consta- cyclic codes overF q2 +vF q2. Quantum Information Processing. 20(3), 122 (2021)

    2021

  8. [8]

    Discrete Mathematics

    Gao, J., Zhang, Y ., Liu, Y ., et al.: New MDS EAQECCs derived from constacyclic codes overF q2 +vF q2. Discrete Mathematics. 346(9), 113513 (2023)

    2023

  9. [9]

    Quantum Information Processing

    Tian, Z., Gao, J., Gao, Y .: Hulls of constacyclic codes over finite non-chain rings and their applications in quantum codes construction. Quantum Information Processing. 23(1), 9 (2024)

    2024

  10. [10]

    Quantum Information Processing

    Pandey, O.P., Pathak, S., Shukla, A.K., et al.: A study of QECCs and EAQECCs construc- tion from cyclic codes over the ringF q +v 1Fq +v 2Fq +· · ·+v sFq. Quantum Information Processing. 23(2), 31 (2024)

    2024

  11. [11]

    Computational and Applied Mathematics

    Yadav, S., Singh, A., Islam, H., et al.: Hermitian hull of constacyclic codes over a class of non-chain rings and new quantum codes. Computational and Applied Mathematics. 43(5), 269 (2024)

    2024

  12. [12]

    Quantum Information Processing

    Zhang, X., Hu, P.: New QEC codes from cyclic codes over finite chain rings. Quantum Information Processing. 24(8), 254 (2025)

    2025

  13. [13]

    Cryptography and Communications

    Wang, Y ., Kai, X., Sun, Z., Zhu, S.: Hermitian dual-containing constacyclic codes over Fq2 +v 1Fq2 +· · ·+v rFq2 and new quantum codes. Cryptography and Communications. 15(1), 145-158 (2023)

    2023

  14. [14]

    Quantum Information Processing

    Zhang, Y ., Liu, Y ., Hou, X., et al.: New MDS operator quantum error-correcting codes derived from constacyclic codes overF q2 +vF q2. Quantum Information Processing. 22(6), 247 (2023)

    2023

  15. [15]

    Quantum Information Processing

    Li, H., Liu, X.: Cyclic codes over a semi-local ring and their applications to QEC and EAQEC codes. Quantum Information Processing. 24(2), 49 (2025)

    2025

  16. [16]

    Quantum Infor- mation Processing

    Dinh, H.Q., Pathak, S., Bag, T., Upadhyay, A.K., Yamaka, W.: Constacyclic codes over mixed alphabets and their applications in constructing new quantum codes. Quantum Infor- mation Processing. 20(4), 150 (2021)

    2021

  17. [17]

    International Journal of Theoretical Physics

    C ¸ alıs ¸kan F., Yıldırım, T., Aksoy, R.: Non-binary quantum codes from cyclic codes over Fp ×(F p +vF p). International Journal of Theoretical Physics. 62(2), 29 (2023) 22

    2023

  18. [18]

    Quantum Information Processing

    Wang, L., Zhang, X., Zhu, S.: Constacyclic codes overF q2 ×(F q2 +vF q2)and their ap- plications in constructing new quantum codes. Quantum Information Processing. 24(5), 120 (2025)

    2025

  19. [19]

    Journal of Applied Mathematics and Computing

    Qian, Y ., Wang, Y ., Wang, L.: Constacyclic codes overF qRr and new quantum codes. Journal of Applied Mathematics and Computing. 72(5), 158 (2026)

    2026

  20. [20]

    Quantum Information Processing

    Hu, P., Liu, X.: New QEC and EAQEC codes from Euclidean sums and hulls of cyclic codes overF 2 ×(F 2 +vF 2). Quantum Information Processing. 24(6), 179 (2025)

    2025

  21. [21]

    Advances in Mathematics of Communications

    Shukla, A.K., Pandey, O.P., Mishra, V ., et al.: Hulls ofZ pZp[v]-cyclic codes and construc- tion of EAQECCs. Advances in Mathematics of Communications. 20, 209-238 (2026)

    2026

  22. [22]

    Quantum Information Processing

    Li, J., Gao, J., Fu, F.W., Ma, F.:F qR-linear skew constacyclic codes and their application of constructing quantum codes. Quantum Information Processing. 19(7), 193 (2020)

    2020

  23. [23]

    Journal of Applied Mathematics and Computing

    Dinh, H.Q., Bag, T., Kewat, P.K., Pathak,S., et al.: Constacyclic codes of length(p r,p s) over mixed alphabets. Journal of Applied Mathematics and Computing. 67(1), 807-832 (2021)

    2021

  24. [24]

    Quantum Information Processing

    Ashraf, M., Khan, N., Mohammad, G.: Quantum codes from cyclic codes over the mixed alphabet structure. Quantum Information Processing. 21(5), 180 (2022)

    2022

  25. [25]

    Designs, Codes and Cryptography

    Ali, S., Aydin, N., Sharma, P., et al.: Constacyclic codes over general mixed alphabets and their applications. Designs, Codes and Cryptography. 93(9), 3733-3760 (2025)

    2025

  26. [26]

    Computational and Applied Mathematics

    Prakash, O., Patel, S., Islam, H.: Quantum codes from skew cyclic codes over mixed alpha- bets. Computational and Applied Mathematics. 44(8), 1-21 (2025)

    2025

  27. [27]

    Designs, Codes and Cryptography

    Fan, Y ., Zhang, L.: Galois self-dual constacyclic codes. Designs, Codes and Cryptography. 84(3), 473-492 (2017)

    2017

  28. [28]

    Discrete Mathematics

    Zhu, S., Wang, L.: A class of constacyclic codes overFp +vF p and its Gray image. Discrete Mathematics. 311(23-24), 2677-2682 (2011)

    2011

  29. [29]

    North-Holland

    MacWilliams, F., Sloane, N.: The theory of error-correcting codes. North-Holland. (1977)

    1977

  30. [30]

    Finite Fields and Their Applications

    Sangwisut, E., Jitman, S., Ling, S., Udomkavanich, P.: Hulls of cyclic and negacyclic codes over finite fields. Finite Fields and Their Applications. 33, 232-257 (2015)

    2015

  31. [31]

    Advances in Mathematics of Communications

    Dougherty, S.T., Sahinkaya, S.: On cyclic and negacyclic codes with one-dimensional hulls and their applications. Advances in Mathematics of Communications. 18(5), 1364- 1378 (2024)

    2024

  32. [32]

    Quantum Information Processing

    Zhang, X.: Good rate QECCs from the quantum construction X. Quantum Information Processing. 22(1), 1 (2023)

    2023

  33. [33]

    Edel, Y .: Some good quantum twisted codes [EB/OL].https://www.yvesedel.de/ Matritzen/QTBCH/QTBCHIndex.html(2025) 23

    2025