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arxiv: 2601.03165 · v2 · submitted 2026-01-06 · 💻 cs.IT · math.IT

On the Euclidean duals of the cyclic codes generated via cyclotomic polynomials

Anuj Kumar Bhagat , Ritumoni Sarma This is my paper

Pith reviewed 2026-05-16 17:01 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords cyclic codescyclotomic polynomialsEuclidean dualminimum distancefinite fieldserror-correcting codesconjecture resolution
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The pith

The minimum distance of the Euclidean duals of cyclic codes generated by the n-th cyclotomic polynomial is exactly 2 raised to the number of distinct prime factors of n.

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

The paper determines the full structure of the Euclidean dual codes of two families of cyclic codes of length n over finite fields F_q. One family is generated by the n-th cyclotomic polynomial Q_n(x) and the other by the product Q_n(x) Q_1(x). From this structure it derives that both duals have minimum distance equal to 2^ω(n), where ω(n) counts the distinct prime factors of n. A reader cares because the distance fixes the exact number of errors these dual codes can detect or correct, and the result settles a prior conjecture on the same codes.

Core claim

We completely describe the structure of the Euclidean duals of the cyclic codes C_n and C_{n,1} and thereby prove that the minimum distance of each dual equals 2^ω(n). The result holds whenever n is at least 2 and coprime to the characteristic of F_q.

What carries the argument

The generator polynomials Q_n(x) and Q_n(x)Q_1(x) together with the cyclotomic factorization of x^n-1 that determines the dual generator polynomials.

If this is right

  • Both dual codes have minimum distance exactly 2^ω(n).
  • When n is prime the dual distance is exactly 2.
  • The dual codes are themselves cyclic with generators that can be written explicitly from the original cyclotomic factors.
  • The error-correcting capability of each dual is known for every admissible n.

Where Pith is reading between the lines

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

  • For square-free n the distance grows only with the number of prime factors, not with the size of n itself.
  • The same structural description may extend to other divisors of x^n-1 that are products of distinct cyclotomic polynomials.
  • Explicit dual distances make it straightforward to test whether these codes are suitable for constructing self-orthogonal or quantum codes.

Load-bearing premise

The field characteristic does not divide n, so the roots of the cyclotomic polynomials behave as expected in the splitting field.

What would settle it

For n=6 over F_5 compute the actual minimum distance of the dual of C_6; if it is not exactly 4 then the claim fails.

read the original abstract

For a natural number $n\ge2$ which is co-prime to Char$(\mathbb{F}_q)$, let $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ denote the cyclic codes of length $n$ over $\mathbb{F}_q$ generated by the $n$-th cyclotomic polynomial $Q_n(x)$ and the polynomial $Q_n(x)Q_1(x)$, respectively. In \cite{BHAGAT2025}, the minimum distances of the codes $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ were determined, and a conjecture regarding the minimum distances of their Euclidean duals was proposed. In this article, we completely describe the structure of these dual codes and as a consequence, we find their minimum distances explicitly as functions of $n$. In fact, we resolve the conjecture in \cite{BHAGAT2025} by proving that the minimum distance of the Euclidean dual of each of $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ is equal to $2^{\omega(n)}$.

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 / 1 minor

Summary. The paper defines cyclic codes C_n and C_{n,1} of length n over F_q (n coprime to char(F_q)) generated by the cyclotomic polynomial Q_n(x) and by Q_n(x)Q_1(x), respectively. It gives an explicit algebraic description of the Euclidean duals of these codes via the reciprocal polynomials of the cyclotomic factors and proves that both duals have minimum distance exactly 2^{ω(n)}, thereby resolving the conjecture stated in BHAGAT2025.

Significance. The work supplies a complete, parameter-free description of the dual codes together with closed-form minimum-distance expressions that depend only on the number of distinct prime factors of n. The proof rests on the standard factorization of x^n-1, the definition of the Euclidean dual, and a direct weight analysis of the idempotent generators; it therefore strengthens the structural theory of cyclotomic cyclic codes without introducing new assumptions or computational fitting.

minor comments (1)
  1. [§2] §2: the notation for the reciprocal polynomial of Q_n(x) is introduced without an explicit equation label; adding a numbered display would improve traceability when the dual generator is later referenced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive recommendation to accept. The referee's summary correctly identifies the main results: the explicit algebraic description of the Euclidean duals of C_n and C_{n,1} together with the closed-form minimum-distance formula 2^{ω(n)} that resolves the conjecture from BHAGAT2025.

Circularity Check

0 steps flagged

No significant circularity; standard algebraic derivation

full rationale

The derivation proceeds from the standard factorization x^n-1 = product_{d|n} Q_d(x) over F_q (n coprime to q) and the definition of the Euclidean dual as the code generated by the reciprocal polynomial. The minimum-distance result 2^ω(n) is obtained by counting the number of distinct prime factors in the support of the idempotent generators of the dual; this counting uses only the divisor lattice of n and the BCH bound applied to the dual roots. The single self-citation to the conjecture in [BHAGAT2025] is purely motivational and is not invoked as a theorem inside the proof. No fitted parameters, self-definitional loops, or ansätze imported from prior work appear.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the standard unique factorization of x^n-1 into cyclotomic polynomials over F_q (when gcd(n,q)=1) and the fact that the Euclidean dual of a cyclic code is also cyclic with roots determined by the reciprocal of the original roots. No free parameters or new entities are introduced.

axioms (2)
  • standard math x^n-1 factors uniquely into distinct cyclotomic polynomials Q_d(x) for d dividing n when gcd(n, char(F_q))=1
    Invoked in the definition of C_n and C_{n,1} and in locating the roots of the dual generator polynomials.
  • standard math The Euclidean dual of a cyclic code generated by g(x) is the cyclic code generated by the reciprocal polynomial of (x^n-1)/g(x)
    Used to obtain the explicit generator of the dual from the roots of Q_n and Q_1.

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Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    A. Barg, K. Haymaker, E. W. Howe, G. L. Matthews, and A. V´ arilly-Alvarado. Locally recoverable codes from algebraic curves and surfaces. In E. W. Howe, K. E. Lauter, and J. L. Walker, editors,Algebraic Geometry for Coding Theory and Cryptography, pages 95–127, Cham, 2017. Springer International Publishing

    work page 2017

  2. [2]

    A. K. Bhagat and R. Sarma. Cyclic locally recoverable lcd codes with the help of cyclotomic polynomials.Finite Fields and Their Applications, 101:102519, 2025

    work page 2025

  3. [3]

    V. R. Cadambe and A. Mazumdar. Bounds on the size of locally recoverable codes.IEEE Trans. Inform. Theory, 61(11):5787–5794, 2015

    work page 2015

  4. [4]

    Carlet and S

    C. Carlet and S. Guilley. Complementary dual codes for counter-measures to side-channel attacks.Adv. Math. Commun., 10(1):131–150, 2016

    work page 2016

  5. [5]

    Gopalan, C

    P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin. On the locality of codeword symbols. IEEE Trans. Inform. Theory, 58(11):6925–6934, 2012. 10

    work page 2012

  6. [6]

    L. Jin, L. Ma, and C. Xing. Construction of optimal locally repairable codes via auto- morphism groups of rational function fields.IEEE Transactions on Information Theory, 66(1):210–221, 2020

    work page 2020

  7. [7]

    F. Li, H. Chen, and S. Lyu. A characterization of optimal locally repairable codes. Discrete Math., 346(7):Paper No. 113465, 8, 2023

    work page 2023

  8. [8]

    X. Li, L. Ma, and C. Xing. Optimal locally repairable codes via elliptic curves.IEEE Transactions on Information Theory, 65(1):108–117, 2019

    work page 2019

  9. [9]

    Lidl and H

    R. Lidl and H. Niederreiter.Introduction to finite fields and their applications. Cambridge University Press, Cambridge, first edition, 1994

    work page 1994

  10. [10]

    Ling and C

    S. Ling and C. Xing.Coding theory. Cambridge University Press, Cambridge, 2004. A first course

    work page 2004

  11. [11]

    Y. Luo, C. Xing, and C. Yuan. Optimal locally repairable codes of distance 3 and 4 via cyclic codes.IEEE Transactions on Information Theory, 65(2):1048–1053, 2019

    work page 2019

  12. [12]

    F. J. MacWilliams and N. J. A. Sloane.The theory of error-correcting codes, volume 16. Elsevier, 1977

    work page 1977

  13. [13]

    Prange.Cyclic Error-correcting Codes in Two Symbols

    E. Prange.Cyclic Error-correcting Codes in Two Symbols. AFCRC-TN. Air Force Cam- bridge Research Center, 1957

    work page 1957

  14. [14]

    Rajput, M

    C. Rajput, M. Bhaintwal, and R. Bandi. On cyclic lrc codes that are also lcd codes. In2020 5th International Conference on Computing, Communication and Security (IC- CCS), pages 1–5, 2020

    work page 2020

  15. [15]

    Tamo and A

    I. Tamo and A. Barg. A family of optimal locally recoverable codes.IEEE Transactions on Information Theory, 60(8):4661–4676, 2014

    work page 2014

  16. [16]

    P. Tan, Z. Zhou, H. Yan, and U. Parampalli. Optimal cyclic locally repairable codes via cyclotomic polynomials.IEEE Communications Letters, 23(2):202–205, 2019

    work page 2019

  17. [17]

    Zengin and M

    R. Zengin and M. E. K¨ oroˇ glu. Constacyclic locally recoverable codes from their duals. Comput. Appl. Math., 43(4):Paper No. 182, 13, 2024

    work page 2024

  18. [18]

    W. Zhao, K. W. Shum, and S. Yang. Optimal locally repairable constacyclic codes of prime power lengths. In2020 IEEE International Symposium on Information Theory (ISIT), pages 7–12, 2020

    work page 2020

  19. [19]

    W. Zhao, K. W. Shum, and S. Yang. A characterization of optimal constacyclic locally repairable codes.Discrete Mathematics, 347(5):113901, 2024. 11

    work page 2024