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arxiv: 2605.23447 · v1 · pith:IOLCTO2Jnew · submitted 2026-05-22 · 🧮 math.CO

AMDS and quantum AMDS Constacyclic codes of length 4p^varsigma over mathbb{F}_{{p}^(m)}

Manasa K J , Divya Acharya , Prasanna Poojary This is my paper

Pith reviewed 2026-05-25 04:03 UTC · model grok-4.3

classification 🧮 math.CO
keywords constacyclic codesAMDS codesquantum AMDS codesCSS constructionfinite fieldserror-correcting codeslength 4p^ς
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The pith

AMDS constacyclic codes of length 4p^ς over F_{p^m} are constructed explicitly and yield quantum AMDS codes via the CSS framework.

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

The paper examines almost maximum distance separable constacyclic codes whose length is four times a power of an odd prime, over finite fields of the same characteristic. It supplies generator polynomials and defining sets that achieve the AMDS distance bound for these parameters. From suitable self-orthogonal or dual-containing instances of these codes the authors derive quantum AMDS codes using the Calderbank-Shor-Steane construction. A reader would care because the resulting quantum codes offer near-optimal error correction for quantum systems while remaining explicit and parameter-friendly for the stated lengths.

Core claim

The authors establish that constacyclic codes of length 4p^ς over F_{p^m} admit defining sets that produce the AMDS property, and that certain of these codes satisfy the CSS orthogonality condition, thereby producing quantum AMDS codes whose minimum distance reaches one less than the quantum Singleton bound.

What carries the argument

Constacyclic codes whose roots are chosen from cyclotomic cosets so that the resulting minimum distance meets the AMDS bound; the CSS framework then converts dual-containing pairs into quantum codes.

If this is right

  • The classical AMDS codes give efficient error-correcting schemes for channels whose block length matches 4p^ς.
  • The derived quantum codes correct a larger fraction of errors than generic CSS codes at the same length and dimension.
  • The constructions remain algebraic and therefore admit fast encoding and decoding algorithms based on cyclotomic coset structure.
  • Parameters can be scaled by increasing ς or m without leaving the AMDS regime.

Where Pith is reading between the lines

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

  • The same length family may admit further MDS or near-MDS constacyclic codes when the multiplier is chosen differently from the standard constacyclic case.
  • The quantum codes could be concatenated with existing quantum LDPC constructions to reach larger block lengths while preserving the AMDS distance scaling.
  • The explicit generator polynomials allow direct verification of the distance bound by computer for small p and ς, providing a practical check on the general construction.

Load-bearing premise

That explicit defining sets exist which simultaneously enforce the required distance for AMDS and the dual-containment condition for the CSS construction at every length 4p^ς.

What would settle it

An explicit computation for the smallest case (p=3, ς=1, m=1) showing that no constacyclic code of length 12 over F_3 meets both the AMDS distance and the CSS dual-containment condition simultaneously.

read the original abstract

This paper provides a comprehensive analysis of almost maximum distance separable (AMDS) constacyclic codes of length $4p^{\varsigma}$ over the finite field $\mathbb{F}_{p^m}$, where $p$ is an odd prime number. Furthermore, it introduces the construction of quantum AMDS (qAMDS) codes, drawing on the principles of the Calderbank-Shor-Steane (CSS) framework, which enhances their applicability in quantum error correction. This work aims to deepen the understanding of these codes and their potential uses in both classical and quantum computing environments.

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

0 major / 2 minor

Summary. The manuscript analyzes almost maximum distance separable (AMDS) constacyclic codes of length 4p^ς over F_{p^m} (p odd prime) and constructs quantum AMDS codes from them via the CSS framework, claiming explicit constructions that achieve the AMDS distance bound d = n - k and satisfy the required orthogonality conditions.

Significance. If the claimed explicit constructions via defining sets hold, the work supplies new families of constacyclic AMDS codes and their quantum lifts for a specific length class, which can be useful for quantum error correction when the parameters are competitive with existing tables.

minor comments (2)
  1. The abstract and introduction should explicitly state the main theorems (e.g., the precise form of the defining sets that yield the AMDS property) rather than only describing the existence of an analysis.
  2. Notation for the length parameter ς and the field extension degree m should be introduced with a clear table of admissible ranges before the first theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The manuscript presents explicit constructions of AMDS constacyclic codes of length 4p^ς over F_{p^m} (p odd prime) together with their quantum AMDS lifts via the CSS construction. No major comments appear in the report, so there are no individual points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs AMDS constacyclic codes of length 4p^ς via explicit defining sets over F_{p^m} and lifts them to qAMDS codes through the external CSS framework. No step reduces a claimed prediction or distance bound to a fitted parameter by construction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The central claims rest on parameter choices and generator polynomials whose independence from the target distance is stated directly, rendering the chain non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities can be identified from the abstract alone.

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Semisimple Cyclic and Abelian Codes II,Cybernetics, 3(3), 17–23, 1967

    SD Berman. Semisimple Cyclic and Abelian Codes II,Cybernetics, 3(3), 17–23, 1967

    work page 1967

  2. [2]

    JL Massey, DJ Costello Jr, and J Justesen, Polynomial Weights and Code Constructions, IEEE Transactions on Information Theory, 19(1), 101–110, 1973

    work page 1973

  3. [3]

    RM Roth and G Seroussi, On Cyclic MDS Codes of LengthqoverGF(q),IEEE transactions on information theory, 32(2), 284–285, 1986

    work page 1986

  4. [4]

    Jacobus H van Lint, Repeated-root cyclic codes,IEEE Transactions on Information Theory, 37(2), 343–345, 1991

    work page 1991

  5. [5]

    On Repeated-Root Cyclic Codes,IEEE Transactions on Information Theory, 37(2), 337– 342, 1991

    Guy Castagnoli, James L Massey, Philipp A Schoeller, and Niklaus Von Seemann. On Repeated-Root Cyclic Codes,IEEE Transactions on Information Theory, 37(2), 337– 342, 1991

    work page 1991

  6. [6]

    Florence Jessie MacWilliams and Neil James Alexander Sloane,The Theory of Error- Correcting Codes, Elsevier, 1977

    work page 1977

  7. [7]

    Almost MDS Codes,Designs, Codes and Cryptography, 9(2), 143–155, 1996

    Mario A De Boer. Almost MDS Codes,Designs, Codes and Cryptography, 9(2), 143–155, 1996

    work page 1996

  8. [8]

    On Near-MDS Codes.Journal of Geometry, 54(1-2), 30–43, 1995

    Stefan Dodunekov and Ivan Landgev. On Near-MDS Codes.Journal of Geometry, 54(1-2), 30–43, 1995

    work page 1995

  9. [9]

    Dodunekov, and Torleiv Klove, Almost-MDS and Near-MDS Codes for Error Detection,IEEE Transactions on Information Theory, 43(1), 285–290, 2002

    Rossitza Dodunekova, Stefan M. Dodunekov, and Torleiv Klove, Almost-MDS and Near-MDS Codes for Error Detection,IEEE Transactions on Information Theory, 43(1), 285–290, 2002

    work page 2002

  10. [10]

    Andreas Faldum and Wolfgang Willems, Codes of Small Defect.Designs, Codes and Cryptography, 10(3), 341–350, 1997

    work page 1997

  11. [11]

    Andreas Faldum and Wolfgang Willems, A characterization of Codes with Extreme Parameters.IEEE Transactions on Information Theory, 42(6), 2255–2257, 2002

    work page 2002

  12. [12]

    Xiaojun Geng, Meng Yang, Jun Zhang, and Zhengchun Zhou, A class of Almost MDS Codes,Finite Fields and Their Applications, 79, 101996, 2022. 16

    work page 2022

  13. [13]

    Quantum Information Processing, 20(11), 373, 2021

    Hai Q Dinh, Ha T Le, Bac T Nguyen, and Roengchai Tansuchat, Quantum MDS and Synchronizable Codes from Cyclic and Negacyclic Codes of Length 4p s overF pm. Quantum Information Processing, 20(11), 373, 2021

    work page 2021

  14. [14]

    Saroj Rani, Ram Krishna Verma, and Om Prakash, Quantum Codes from Repeated- root Cyclic and Negacyclic Codes of Length 4p s overF pm.International Journal of Theoretical Physics, 60(4), 1299–1327, 2021

    work page 2021

  15. [15]

    Peter W Shor, Scheme for reducing decoherence in quantum computer memory.Physical review A, 52(4), R2493, 1995

    work page 1995

  16. [16]

    Nested Quantum Error Correction Codes,IEEE Trans

    A Calderbank, E Rains, P Shor, and NJA Sloane. Nested Quantum Error Correction Codes,IEEE Trans. Inf. Theory, 44(4), 1369–1387, 1998

    work page 1998

  17. [17]

    Hai Q Dinh, Bac T Nguyen, and Hiep L Thi, AMDS Constacyclic Codes and Quantum AMDS Codes.Filomat, 38(33), 11889–11912, 2024

    work page 2024

  18. [18]

    Hai Q Dinh, Bac T Nguyen, Nui V Nguyen, Hiep L Thi, and Woraphon Yamaka, Quantum AMDS Negacyclic Codes of Length 2p s overF pm .IEEE Access, 2025

    work page 2025

  19. [19]

    Hai Q. Dinh, Xiaoqiang Wang, Hongwei Liu, Songsak Sriboonchitta, On the Ham- ming distances of repeated-root constacyclic codes of length 4p s, Discrete Mathematics, 342(5), 1456-1470, 2019

    work page 2019

  20. [20]

    Hai Q Dinh, Bac T Nguyen, and Hiep L Thi, AMDS Codes and Quantum AMDS Codes of Length 3p s overF pm,Applicable Algebra in Engineering, Communication and Computing, 1–18, 2025

    work page 2025

  21. [21]

    Constacyclic Codes of Lengthp s overF pm +uF pm.Journal of Algebra, 324(5), 940–950, 2010

    Hai Q Dinh. Constacyclic Codes of Lengthp s overF pm +uF pm.Journal of Algebra, 324(5), 940–950, 2010

    work page 2010

  22. [22]

    On Repeated-Root Constacyclic Codes of Length 4p s,Asian-European Journal of Mathematics, 6(2), 1350020, 2013

    Hai Q Dinh. On Repeated-Root Constacyclic Codes of Length 4p s,Asian-European Journal of Mathematics, 6(2), 1350020, 2013

    work page 2013

  23. [23]

    Good Quantum Error-Correcting Codes Exist, Physical Review A, 54(2), 1098–1105, 1996

    AR Calderbank and Peter W Shor. Good Quantum Error-Correcting Codes Exist, Physical Review A, 54(2), 1098–1105, 1996

    work page 1996

  24. [24]

    Markus Grassl, Thomas Beth, and Martin R¨ otteler, On optimal quantum codes,Inter- national Journal of Quantum Information, 2(1), 55–64, 2004

    work page 2004

  25. [25]

    RAINS, EM, Quantum weight enumerators, IEEE Trans. Inform. Theory, 44(4), 1388– 1394, 1998

    work page 1998

  26. [26]

    Roman, Steven Coding and information theory, Springer Science & Business Media 134, 1992. 17

    work page 1992

  27. [27]

    Calderbank, AR and Shor, Peter W Good quantum error-correcting codes exist, Calder- bank, AR and Shor, Peter W, PHYSICAL REVIEW A, 54(2), 1996 18

    work page 1996