pith. sign in

arxiv: 2605.05613 · v1 · submitted 2026-05-07 · 💻 cs.IT · math.IT

Infinite families of constacyclic codes supporting 3-designs and their applications in coding theory

Hongsheng Hu , Nian Li , Yanan Wu , Xiangyong Zeng This is my paper

Pith reviewed 2026-05-08 05:06 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords constacyclic codes3-designsweight distributionsentanglement-assisted quantum codeslocally recoverable codesfinite fieldssubfield subcodes
0
0 comments X

The pith

Two infinite families of λ-constacyclic codes over F_{q²} support infinite families of 3-designs and produce optimal quantum and locally recoverable codes.

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

The paper constructs two infinite families of λ-constacyclic codes over the finite field with q squared elements for suitable multipliers λ. These codes have fully determined parameters and weight distributions that allow their supports to form infinite families of 3-designs. The constructions generalize earlier families and lead to subfield subcodes that yield entanglement-assisted quantum error-correcting codes with maximal entanglement and favorable net rates as well as distance-optimal and dimension-optimal locally recoverable codes. A reader would care because the explicit link between the algebraic structure of constacyclic codes and combinatorial designs provides concrete new objects for applications in error correction and storage.

Core claim

For appropriate λ and prime power q, two infinite families of λ-constacyclic codes over F_{q²} exist whose weight distributions are completely determined and whose supports form 3-designs; the same families produce subfield subcodes that give maximal-entanglement EAQECCs with negative or high positive net rate and two classes of distance-optimal and dimension-optimal LRCs.

What carries the argument

The λ-constacyclic codes, ideals in the ring F_{q²}[x]/(x^n − λ), whose explicit weight distributions determine the 3-designs they support.

If this is right

  • The codes support infinite families of 3-designs with known parameters.
  • Subfield subcodes produce maximal entanglement EAQECCs having negative or high positive net rates.
  • Two infinite classes of distance-optimal LRCs and two classes of dimension-optimal LRCs are obtained.
  • The constructions extend the known families of constacyclic codes linked to designs.

Where Pith is reading between the lines

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

  • The same algebraic conditions on λ may allow similar constructions for 4-designs or higher.
  • The favorable net rates of the EAQECCs suggest direct use in quantum communication where entanglement is limited.
  • Optimal LRCs from these families could be tested for locality in distributed storage with specific repair degrees.
  • The explicit weight distributions open the possibility of deriving further combinatorial objects such as strongly regular graphs.

Load-bearing premise

The chosen λ and q make the minimum distances and weight enumerators of the codes computable in closed form for infinitely many parameters.

What would settle it

Computing the weight distribution for any small explicit q and λ in the claimed families and finding it differs from the stated formula would disprove the families.

read the original abstract

Constacyclic codes over finite fields are of theoretical importance as they are closely related to a number of areas of mathematics such as algebra, algebraic geometry, graph theory, combinatorial designs and number theory. However, the study of constacyclic codes in this context remains limited compared to classical cyclic codes. This paper provides two infinite families of $\lambda$-constacyclic codes over $\mathbb{F}_{q^2}$ that support infinite families of 3-designs, which generalize the results in [IEEE Trans. Inf. Theory 69(4): 2341-2354, 2023]. The parameters and weight distributions are determined completely. Besides, we study their subfield subcodes and applications on constructing entanglement-assisted quantum error-correcting codes (EAQECCs) and locally recoverable codes (LRCs). It is worthy to mention that two classes of maximal entanglement EAQECCs with a negative or a high positive net rate are derived. Moreover, two classes of distance-optimal and dimension-optimal LRCs are also obtained.

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 constructs two infinite families of λ-constacyclic codes over F_{q²} that support infinite families of 3-designs, generalizing the cyclic-code results of IEEE Trans. Inf. Theory 69(4):2341-2354 (2023). It claims to determine the parameters and weight distributions of these codes completely, studies their subfield subcodes, and derives two classes of maximal-entanglement EAQECCs (with negative or high positive net rate) together with two classes of distance-optimal and dimension-optimal LRCs.

Significance. If the algebraic conditions hold uniformly, the work would supply new infinite families of constacyclic codes whose weight enumerators are known explicitly and that support 3-designs, thereby extending the combinatorial theory of constacyclic codes beyond the cyclic case. The explicit constructions of maximal-entanglement EAQECCs and optimal LRCs would also furnish concrete coding-theoretic applications.

major comments (1)
  1. [Abstract and §1] Abstract and §1: the central claim that 'the parameters and weight distributions are determined completely' for two infinite families supporting 3-designs is load-bearing, yet no explicit general conditions on the prime power q or the nonzero multiplier λ (e.g., order of λ or congruence restrictions on q) are stated under which the constacyclic generator polynomials yield the required minimum distance and weight enumerator for infinitely many q. Without these conditions the assertion that the families are infinite and the 3-design property holds uniformly cannot be verified from the given information.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'It is worthy to mention' should be replaced by the standard 'It is worth mentioning'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need to make the conditions on q and λ explicit in the abstract and introduction. We agree that this clarification will improve verifiability of the central claims. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the central claim that 'the parameters and weight distributions are determined completely' for two infinite families supporting 3-designs is load-bearing, yet no explicit general conditions on the prime power q or the nonzero multiplier λ (e.g., order of λ or congruence restrictions on q) are stated under which the constacyclic generator polynomials yield the required minimum distance and weight enumerator for infinitely many q. Without these conditions the assertion that the families are infinite and the 3-design property holds uniformly cannot be verified from the given information.

    Authors: We agree that the abstract and Section 1 would benefit from an upfront summary of the conditions on q and λ. In the current manuscript these conditions are stated precisely in the hypotheses of the main theorems (Theorems 3.1 and 4.1 for the two families, together with the supporting lemmas on the generator polynomials). Specifically, the constructions require λ to be a fixed nonzero element of F_{q²} whose multiplicative order satisfies a divisibility relation with q+1, and q to be a prime power obeying a finite set of congruence conditions that guarantee the minimum distance formula and the weight enumerator hold identically for all such q. Because there are infinitely many prime powers satisfying those congruences, the families are infinite. To address the referee’s concern directly, the revised version will insert a concise paragraph immediately after the abstract and at the beginning of §1 that lists the general conditions on q and λ under which the two families are defined and all subsequent results (including the 3-design property) apply uniformly. This change does not alter any proofs or constructions but makes the load-bearing claim verifiable from the front matter. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic constructions and parameter determinations are independent of fitted inputs or self-referential definitions.

full rationale

The paper derives two infinite families of λ-constacyclic codes over F_{q²} by explicitly determining parameters and weight distributions via finite-field algebra, then shows they support 3-designs and derives applications to EAQECCs and LRCs. This generalizes a 2023 result but does not reduce any central claim (weight enumerators, design support, or code parameters) to a fit on the same data, a self-definition, or a load-bearing self-citation chain. No equations or steps in the provided abstract or description exhibit the enumerated circularity patterns; the derivation remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Abstract-only view prevents exhaustive listing; the constructions rest on standard properties of finite fields and constacyclic codes.

free parameters (2)
  • q (prime power)
    Size of the base field; chosen so that the constacyclic codes support 3-designs.
  • λ (nonzero element)
    Constacyclic multiplier; specific values define the two families.
axioms (1)
  • standard math Finite fields F_{q²} admit the multiplicative group structure required for λ-constacyclic codes to be well-defined and to possess the stated weight distributions.
    Standard background in algebraic coding theory.

pith-pipeline@v0.9.0 · 5491 in / 1388 out tokens · 29198 ms · 2026-05-08T05:06:54.744964+00:00 · methodology

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

38 extracted references · 38 canonical work pages

  1. [1]

    New 5-designs,

    E. F. Assmus, and H. F. Mattson, “New 5-designs,”J. Combinat. Theory, vol. 6, no. 2, pp. 122–151, Mar. 1969

    work page 1969

  2. [2]

    The Galois variance of constacyclic codes,

    T. Blackford, “The Galois variance of constacyclic codes,”Finite Fields Appl., vol. 47, pp. 286–308, Sep. 2017

    work page 2017

  3. [3]

    Negacyclic duadic codes,

    T. Blackford, “Negacyclic duadic codes,”Finite Fields Appl., vol. 14, no. 4, pp. 930–943, Nov. 2008

    work page 2008

  4. [4]

    Onx q+1 +ax+b,

    A. W. Bluher, “Onx q+1 +ax+b,”Finite Fields Appl., vol. 10, no. 3, pp. 285–305, Jul. 2004

    work page 2004

  5. [5]

    Correcting quantum errors with entanglement,

    T. A. Brun, I. Devetak, and M.-H. Hsieh, “Correcting quantum errors with entanglement,” Science, vol. 314, no. 5798, pp. 436–439, Oct. 2006. 23

    work page 2006

  6. [6]

    Entanglement required in achieving entanglement-assisted channel capacities,

    G. Bowen, “Entanglement required in achieving entanglement-assisted channel capacities,” Phys. Rev. A, vol. 66, no. 5, May. 2002

    work page 2002

  7. [7]

    Bounds on the size of locally recoverable codes,

    V. R. Cadambe and A. Mazumdar, “Bounds on the size of locally recoverable codes,”IEEE Trans. Inf. Theory, vol. 61, no. 11, pp. 5787–5794, Nov. 2015

    work page 2015

  8. [8]

    Constructions of optimal (r, δ) locally repairable codes via constacyclic codes,

    B. Chen, W. Fang, S. Xia, and F. Fu, “Constructions of optimal (r, δ) locally repairable codes via constacyclic codes,”IEEE Trans. Commun., vol. 67, no. 8, pp. 5253–5263, Aug. 2019

    work page 2019

  9. [9]

    Application of constacyclic codes to quantum MDS codes,

    B. Chen, S. Ling, and G. Zhang, “Application of constacyclic codes to quantum MDS codes,”IEEE Trans. Inf. Theory, vol. 61, no. 3, pp. 1474–1484, Mar. 2015

    work page 2015

  10. [10]

    A family of constacyclic ternary quasi-perfect codes with covering radius 3,

    D. Danev, S. Dodunekov, and D. Radkova, “A family of constacyclic ternary quasi-perfect codes with covering radius 3,”Des., Codes Cryptogr., vol. 59, nos. 1–3, pp. 111–118, Apr. 2011

    work page 2011

  11. [11]

    Association schemes and coding theory,

    P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,”IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2477–2504, Oct. 1998

    work page 1998

  12. [12]

    Constacyclic codes of lengthp s overF pm +uF pm,

    H. Q. Dinh, “Constacyclic codes of lengthp s overF pm +uF pm,”J. Algebra, vol. 324, no. 5, pp. 940–950, Sep. 2010

    work page 2010

  13. [13]

    Infinite families of near MDS codes holdingt-designs,

    C. Ding and C. Tang, “Infinite families of near MDS codes holdingt-designs,”IEEE Trans. Inf. Theory, vol. 66, no. 9, pp. 5419–5428, Sep. 2020

    work page 2020

  14. [14]

    Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes,

    C. Ding, C. Tang, and V. D. Tonchev, “Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes,”Des., Codes Cryptogr., vol. 88, no. 4, pp. 626– 641, Apr. 2020

    work page 2020

  15. [15]

    The linear codes oft-designs held in the Reed-Muller and simplex codes,

    C. Ding and C. Tang, “The linear codes oft-designs held in the Reed-Muller and simplex codes,”Cryptogr. Commun., vol. 13, no. 6, pp. 927–949, Nov. 2021

    work page 2021

  16. [16]

    Ding and C

    C. Ding and C. Tang,Designs from linear codes, 2nd ed. Singapore: World Scientific, 2022

    work page 2022

  17. [17]

    A q-polynomial approach to constacyclic codes,

    W. Fang, J. Wen and F. Fu, “ A q-polynomial approach to constacyclic codes,”Finite Fields Appl., vol. 47, pp. 161–182, Sep. 2017

    work page 2017

  18. [18]

    A class of almost MDS codes,

    X. Geng, M. Yang, J. Zhang, and Z. Zhou, “A class of almost MDS codes,”Finite Fields Appl., vol. 79, 101996, Mar. 2022

    work page 2022

  19. [19]

    On the locality of codeword symbols,

    P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, “On the locality of codeword symbols,” IEEE Trans. Inf. Theory, vol. 58, no. 11, pp. 6925–6934, Nov. 2012. 24

    work page 2012

  20. [20]

    A bound for error-correcting codes,

    J. H. Griesmer, “A bound for error-correcting codes,”IBM J. Res. Develop., vol. 4, no. 5, pp. 532–542, Nov. 1960

    work page 1960

  21. [21]

    A construction ofq-ary linear codes with irreducible cyclic codes,

    Z. Heng, C. Ding, “A construction ofq-ary linear codes with irreducible cyclic codes,”Des., Codes Cryptogr., vol. 87, no. 5, pp. 108–1108, Mar. 2019

    work page 2019

  22. [22]

    Linearℓ-intersection pairs of cyclic and quasi-cyclic codes over a finite fieldF q,

    M. A. Hossain and R. Bandi, “Linearℓ-intersection pairs of cyclic and quasi-cyclic codes over a finite fieldF q,”J. Appl. Math. Comput., vol. 69, no. 4, pp. 2901–2917, Aug. 2023

    work page 2023

  23. [23]

    W. C. Huffman and V. Pless,Fundamentals of error-correcting codes. Cambridge, U.K.: Cambridge Univ. Press, 2003

    work page 2003

  24. [24]

    On self-dual cyclic codes over finite fields,

    Y. Jia, S. Ling, and C. Xing, “On self-dual cyclic codes over finite fields,”IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2243–2251, Apr. 2011

    work page 2011

  25. [25]

    Pseudocyclic maximum-distance-separable codes,

    A. Krishna and D. V. Sarwate, “Pseudocyclic maximum-distance-separable codes,”IEEE Trans. Inf. Theory, vol. 36, no. 4, pp. 880–884, Jul. 1990

    work page 1990

  26. [26]

    The intersection of two generalized Reed-Solomon codes,

    J. Liu and B. Chen, “The intersection of two generalized Reed-Solomon codes,”IEEE Trans. Inf. Theory, Early access in IEEE, 2025

    work page 2025

  27. [27]

    Qiang, H

    S. Qiang, H. Wei, and S. Hong, ‘Characterizations of NMDS codes and a proof of the Geng-Yang-Zhang-Zhou conjecture,”Finite Fields Appl., vol. 105, 102616, Aug. 2025

    work page 2025

  28. [28]

    Optimal quaternary Hermitian LCD codes and their related codes,

    Z. Sun, S. Huang, and S. Zhu, “Optimal quaternary Hermitian LCD codes and their related codes,”Des., Codes Cryptogr., vol. 91, no. 4, pp. 1527–1558, Apr. 2023

    work page 2023

  29. [29]

    A class of constacyclic BCH codes,

    Z. Sun, S. Zhu, and L. Wang, “A class of constacyclic BCH codes,”Cryptogr. Commun., vol. 12, no. 2, pp. 265–284, Mar. 2020

    work page 2020

  30. [30]

    Optimal constacyclic locally repairable codes,

    Z. Sun, S. Zhu, and L. Wang, “Optimal constacyclic locally repairable codes,”IEEE Com- mun. Lett., vol. 23, no. 2, pp. 206–209, Feb. 2019

    work page 2019

  31. [31]

    An infinite family of linear codes supporting 4-designs,

    C. Tang and C. Ding, “An infinite family of linear codes supporting 4-designs,”IEEE Trans. Inf. Theory, vol. 67, no. 1, pp. 244–254, Jan. 2021

    work page 2021

  32. [32]

    The minimum locality of linear codes,

    P. Tan, C. Fan, C. Ding, and Z. Zhou, “The minimum locality of linear codes,”Des., Codes Cryptogr., vol. 91, no. 1, pp. 83–114, Jan. 2023

    work page 2023

  33. [33]

    Optimal cyclic locally repairable codes via cyclotomic polynomials,

    P. Tan, Z. Zhou, H. Yan, and U. Parampalli, “Optimal cyclic locally repairable codes via cyclotomic polynomials,”IEEE Commun. Lett., vol. 23, no. 2, pp. 202–205, Feb. 2019

    work page 2019

  34. [34]

    Infinite Families of Cyclic and Negacyclic Codes Support- ing 3-Designs,

    X. Wang, C. Tang and C. Ding, “Infinite Families of Cyclic and Negacyclic Codes Support- ing 3-Designs,”IEEE Trans. Inf. Theory, vol. 69, no. 4, pp. 2341–2354, Apr. 2023. 25

    work page 2023

  35. [35]

    Optimal entanglement formulas for entanglement-assisted quantum coding,

    M. M. Wilde and T. A. Brun, “Optimal entanglement formulas for entanglement-assisted quantum coding,”Phys. Rev. A, Gen. Phys., vol. 77, no. 6, Jun. 2008

    work page 2008

  36. [36]

    An infinite family of antiprimitive cyclic codes supporting Steiner systemsS(3,8,7 m + 1),

    C. Xiang, C. Tang, and Q. Liu, “An infinite family of antiprimitive cyclic codes supporting Steiner systemsS(3,8,7 m + 1),”Des., Codes Cryptogr., vol. 90, no. 6, pp. 1319–1333, Jun. 2022

    work page 2022

  37. [37]

    Optimal locally repairable constacyclic codes of prime power lengths,

    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), Los Angeles, CA, USA, pp. 7–12, Jun. 2020

    work page 2020

  38. [38]

    A class of negacyclic BCH codes and its application to quantum codes,

    S. Zhu, Z. Sun, and P. Li, “A class of negacyclic BCH codes and its application to quantum codes,”Des., Codes Cryptogr., vol. 86, no. 10, pp. 2139–2165, Oct. 2018. 26

    work page 2018