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arxiv: 2605.14603 · v1 · pith:IDJ2TRRVnew · submitted 2026-05-14 · 💻 cs.IT · math.IT

Quaternary codes with new parameters from two-generator simplicial complexes

Ankit Yadav , Nilay Kumar Mondal , Ritumoni Sarma This is my paper

Pith reviewed 2026-06-30 20:35 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords quaternary codesC_D-codessimplicial complexesLee weight distributionPlotkin boundGriesmer codesminimal codesGray image
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The pith

Two-generator simplicial complexes define infinite families of quaternary C_D-codes whose Lee weight distributions produce new parameters.

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

The paper constructs infinite families of quaternary linear codes C_D over Z_4 by taking defining sets D from two-generator simplicial complexes and explicitly determines their Lee weight distributions. These constructions yield at least 32 codes that improve on or are absent from the current database of best-known quaternary codes, including a Plotkin-optimal family. The authors also give necessary and sufficient conditions for the Gray images of these codes to be linear over F_2, which produces an infinite family of Griesmer codes together with several infinite families of minimal binary linear codes. A reader would care because the method supplies explicit, infinite families of codes whose parameters sit outside the known tables and whose images inherit strong minimality properties.

Core claim

By deriving the defining set D from a two-generator simplicial complex, the associated C_D-code over Z_4 has a Lee weight distribution that can be computed in closed form; when the complex parameters are varied, the resulting codes include at least 32 entries with parameters new or better than those listed in the best-known database, a Plotkin-optimal subfamily, and six projective codes. The same construction admits necessary and sufficient conditions on the complex parameters that make the Gray image linear, thereby producing an infinite family of Griesmer binary codes and multiple infinite families of minimal binary linear codes.

What carries the argument

Two-generator simplicial complex used to produce the defining set D of a C_D-code over Z_4.

If this is right

  • At least 32 quaternary linear codes improve or are absent from the best-known database.
  • A subfamily attains the Plotkin bound.
  • Six of the codes are projective and achieve best-known parameters.
  • The Gray images satisfy linearity conditions that produce an infinite family of Griesmer codes.
  • The same Gray images produce several infinite families of minimal binary linear codes.

Where Pith is reading between the lines

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

  • The projectivity of the six reported codes may allow them to outperform non-projective codes of the same parameters in applications that exploit the projective geometry of the code.
  • The explicit form of the Lee weight distributions may permit direct comparison with other combinatorial constructions of quaternary codes that use different families of simplicial complexes.
  • Because the linearity conditions on the Gray image are necessary and sufficient, the construction supplies a decision procedure for when a given two-generator complex yields a minimal binary code.

Load-bearing premise

The specific two-generator simplicial complexes chosen produce defining sets D whose C_D-codes have Lee weight distributions that are both correctly computed and lie outside the existing database.

What would settle it

Take one of the 32 claimed new codes, recompute its Lee weight enumerator from the given complex parameters, and check whether the resulting length, dimension, and minimum distance match or beat the entry listed for those parameters in the referenced database.

read the original abstract

In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee weight distributions. As a result, we find at least 32 new or improved quaternary linear codes as per the database \cite{aydin2022updated} of best-known quaternary codes, including codes from a Plotkin-optimal family. We also report 6 projective quaternary linear codes with best-known parameters that might outperform the currently reported best-known codes due to their projectivity. Further, we establish necessary and sufficient conditions for their Gray image to be linear, which in turn gives an infinite family of Griesmer codes and several infinite families of minimal binary linear 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

2 major / 2 minor

Summary. The paper constructs infinite families of quaternary linear C_D-codes with defining sets D obtained from two-generator simplicial complexes, explicitly determines their Lee weight distributions, and compares the resulting parameters [n,k,d_L] against the aydin2022updated database of best-known quaternary codes. It reports at least 32 new or improved codes (including a Plotkin-optimal family) and 6 projective codes with best-known parameters, establishes necessary and sufficient conditions for the Gray image to be linear, and derives from this an infinite family of Griesmer codes together with several infinite families of minimal binary linear codes.

Significance. If the Lee weight distributions are correctly computed, the work supplies explicit, infinite families of quaternary codes attaining new or improved parameters, some of which are Plotkin-optimal or projective, together with a direct link via the Gray map to binary Griesmer and minimal codes. The concrete database comparisons and the projectivity observation constitute verifiable contributions that can be checked against existing tables.

major comments (2)
  1. [Weight distribution computation (Section 4)] The central claim of 32 new/improved codes (and the 6 projective ones) rests on the Lee weight distributions computed for the chosen two-generator simplicial complexes; any error in the enumeration formulas that determine the minimum Lee distance d_L would invalidate the novelty assertions relative to the aydin2022updated database. The manuscript should supply at least one fully worked small-parameter example (including the explicit defining set D, the resulting weight enumerator, and the comparison step) so that the derivation can be independently verified.
  2. [Gray-image linearity (Section 5)] The necessary and sufficient conditions for the Gray image to be linear are stated after the weight-distribution results; it is unclear whether these conditions are applied to the same families that produce the 32 new codes or to a separate subfamily, which affects the strength of the claim that the constructions simultaneously yield both new quaternary parameters and new binary Griesmer/minimal codes.
minor comments (2)
  1. [References] The abstract cites aydin2022updated but the reference list entry should be expanded to include the precise title, authors, and version date for reproducibility.
  2. [Results tables] Tables listing the 32 new codes should include the explicit parameters n, k, d_L, the simplicial-complex generators used, and the Lee weight distribution (or at least the minimum distance) so that readers can directly compare with the database.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address each major comment below and will revise the manuscript to incorporate clarifications and additional verification material.

read point-by-point responses

  1. Referee: [Weight distribution computation (Section 4)] The central claim of 32 new/improved codes (and the 6 projective ones) rests on the Lee weight distributions computed for the chosen two-generator simplicial complexes; any error in the enumeration formulas that determine the minimum Lee distance d_L would invalidate the novelty assertions relative to the aydin2022updated database. The manuscript should supply at least one fully worked small-parameter example (including the explicit defining set D, the resulting weight enumerator, and the comparison step) so that the derivation can be independently verified.

    Authors: We agree that an explicit, fully worked small-parameter example would allow independent verification of the weight-distribution formulas. In the revised manuscript we will insert such an example (with concrete D, the full Lee weight enumerator, and the direct comparison against the aydin2022updated database) immediately after the general formulas in Section 4. revision: yes

  2. Referee: [Gray-image linearity (Section 5)] The necessary and sufficient conditions for the Gray image to be linear are stated after the weight-distribution results; it is unclear whether these conditions are applied to the same families that produce the 32 new codes or to a separate subfamily, which affects the strength of the claim that the constructions simultaneously yield both new quaternary parameters and new binary Griesmer/minimal codes.

    Authors: The necessary and sufficient conditions in Section 5 are derived for the same two-generator simplicial-complex families whose Lee weight distributions are computed in Section 4 and which produce the reported new quaternary codes. We will add an explicit sentence at the beginning of Section 5 stating that the linearity conditions apply to these families, thereby clarifying that the quaternary constructions simultaneously yield the new parameters and the binary Griesmer/minimal codes via the Gray map. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs C_D-codes from explicitly chosen two-generator simplicial complexes, derives their Lee weight distributions by direct computation from the defining sets D, and compares the resulting [n,k,d_L] parameters to an external database (aydin2022updated). No step reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the central claims rest on explicit construction and enumeration rather than any input being renamed or forced as output.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5670 in / 933 out tokens · 27958 ms · 2026-06-30T20:35:32.154576+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Construction of codes over a commutative non-unital ring from simplicial complexes and their applications

    cs.IT 2026-06 unverdicted novelty 5.0

    Constructs linear codes over ring S from simplicial complexes, determines parameters of Gray images and subfield-like codes, and derives families of divisible, minimal, and optimal codes with applications to few-weigh...

Reference graph

Works this paper leans on

35 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [1]

    Ashikhmin and A

    A. Ashikhmin and A. Barg. Minimal vectors in linear codes.IEEE Trans. Inform. Theory, 44(5):2010–2017, 1998

    2010

  2. [2]

    An updated database ofZ 4 codes,

    N. Aydin, Y. Lu, and V. R. Onta. An updated database ofZ 4 codes.arXiv preprint arXiv:2208.06832, 2022. Available:http://quantumcodes.info/Z4/

    work page arXiv 2022

  3. [3]

    A. K. Bhagat, R. Sarma, and V. Sagar. Subfield codes ofC D-codes overF 2[x]/⟨x3 −x⟩.Discrete Math., 348(1):Paper No. 114223, 22, 2025. 33

    2025

  4. [4]

    Chabanne, G

    H. Chabanne, G. Cohen, and A. Patey. Towards secure two-party computation from the wire-tap channel. InInformation security and cryptology—ICISC 2013, volume 8565 ofLecture Notes in Comput. Sci., pages 34–46. Springer, Cham, 2014

    2013

  5. [5]

    Chang and J

    S. Chang and J. Y. Hyun. Linear codes from simplicial complexes.Des. Codes Cryptogr., 86(10):2167–2181, 2018

    2018

  6. [6]

    B. Chen, Y. Xu, Z. Hu, N. Li, and X. Zeng. Optimal linear codes with few weights from simplicial complexes.IEEE Trans. Inform. Theory, 71(5):3531–3543, 2025

    2025

  7. [7]

    H. Chen. Griesmer and optimal linear codes from the affine Solomon-Stiffler construction.IEEE Trans. Inform. Theory, 71(9):6834–6843, 2025

    2025

  8. [8]

    Ding and H

    C. Ding and H. Niederreiter. Cyclotomic linear codes of order 3.IEEE Trans. Inform. Theory, 53(6):2274–2277, 2007

    2007

  9. [9]

    A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. Sloane, and P. Sol´ e. TheZ 4-linearity of Kerdock, Preparata, Goethals, and related codes.IEEE Transactions on Information Theory, 40(2):301–319, 1994

    1994

  10. [10]

    Z. Hu, Y. Xu, N. Li, X. Zeng, L. Wang, and X. Tang. New constructions of optimal linear codes from simplicial complexes.IEEE Trans. Inform. Theory, 70(3):1823–1835, 2024

    2024

  11. [11]

    W. C. Huffman and V. Pless.Fundamentals of error-correcting codes. Cambridge University Press, Cambridge, 2003

    2003

  12. [12]

    J. Y. Hyun, J. Jeong, and Y. Lee. Constructing optimal few weight quaternary linear codes via multivariable functions.Cryptogr. Commun., 17(1):57–85, 2025

    2025

  13. [13]

    J. Y. Hyun, J. Lee, and Y. Lee. Infinite families of optimal linear codes constructed from simplicial complexes.IEEE Trans. Inform. Theory, 66(11):6762–6773, 2020

    2020

  14. [14]

    Islam and O

    H. Islam and O. Prakash. NewZ 4 codes from constacyclic codes over a non-chain ring.Comput. Appl. Math., 40(1):Paper No. 12, 17, 2021

    2021

  15. [15]

    X. Meng, J. Gao, and F.-W. Fu. On generalized quasi-cyclic codes overZ 4.Discrete Math., 347(3):Paper No. 113821, 15, 2024

    2024

  16. [16]

    N. K. Mondal and Y. Lee. Optimal binary few-weight codes using a mixed alphabet ring and simplicial complexes.IEEE Trans. Inform. Theory, 70(7):4865–4878, 2024

    2024

  17. [17]

    Sagar and R

    V. Sagar and R. Sarma. Octanary linear codes using simplicial complexes.Cryptogr. Commun., 15(3):599–616, 2023

    2023

  18. [18]

    Sagar and R

    V. Sagar and R. Sarma. Codes over the non-unital non-commutative ringEusing simplicial complexes.IEEE Trans. Inform. Theory, 70(5):3373–3384, 2024

    2024

  19. [19]

    A. Shamir. How to share a secret.Comm. ACM, 22(11):612–613, 1979

    1979

  20. [20]

    Sharma and M

    A. Sharma and M. Bhaintwal. A class of skew-cyclic codes overZ 4 +uZ 4 with derivation.Adv. Math. Commun., 12(4):723–739, 2018. 34

    2018

  21. [21]

    Shi and X

    M. Shi and X. Li. Few-weight codes over a non-chain ring associated with simplicial complexes and their distance optimal Gray image.Finite Fields Appl., 80:Paper No. 101994, 15, 2022

    2022

  22. [22]

    M. Shi, Y. Liu, H. Randriam, L. Sok, and P. Sol´ e. Trace codes overZ 4, and Boolean functions. Des. Codes Cryptogr., 87(6):1447–1455, 2019

    2019

  23. [23]

    M. Shi, L. Qian, L. Sok, N. Aydin, and P. Sol´ e. On constacyclic codes overZ 4[u]/⟨u2 −1⟩and their Gray images.Finite Fields Appl., 45:86–95, 2017

    2017

  24. [24]

    M. Shi, W. Xuan, and P. Sol´ e. Two families of two-weight codes overZ 4.Des. Codes Cryptogr., 88(12):2493–2505, 2020

    2020

  25. [25]

    H. C. Tang and D. Suprijanto. A general family of Plotkin-optimal two-weight codes overZ 4. Des. Codes Cryptogr., 91(5):1737–1750, 2023

    2023

  26. [26]

    Z. X. Wan.Quaternary codes, volume 8 ofSeries on Applied Mathematics. World Scientific Publishing Co., Inc., River Edge, NJ, 1997

    1997

  27. [27]

    Z. Wang, N. Li, X. Zeng, and X. Tang. Trace Codes OverZ 4 and Their Lee Weight Distributions. IEEE Trans. Inform. Theory, 72(2):1051–1066, 2026

    2026

  28. [28]

    Wu and Y

    Y. Wu and Y. Lee. Binary lcd codes and self-orthogonal codes via simplicial complexes.IEEE Communications Letters, 24(6):1159–1162, 2020

    2020

  29. [29]

    Y. Wu, B. Li, W. Fan, and F. Xiao. Two infinite families of quaternary codes.IEEE Trans. Inform. Theory, 70(12):8723–8733, 2024

    2024

  30. [30]

    Y. Wu, C. Li, and J. Y. Hyun. A survey on codes from simplicial complexes.arXiv preprint arXiv:2409.16310, 2024

    work page arXiv 2024

  31. [31]

    Y. Wu, C. Li, L. Zhang, and F. Xiao. Quaternary codes and their binary images.IEEE Trans. Inform. Theory, 70(7):4759–4768, 2024

    2024

  32. [32]

    Y. Wu, X. Zhu, and Q. Yue. Optimal few-weight codes from simplicial complexes.IEEE Trans. Inform. Theory, 66(6):3657–3663, 2020

    2020

  33. [33]

    Yadav, R

    A. Yadav, R. Sarma, and A. K. Bhagat. Optimal binary codes fromC D-codes over a non-chain ring.arXiv preprint arXiv:2510.09057, 2025

    work page arXiv 2025

  34. [34]

    Yuan and C

    J. Yuan and C. Ding. Secret sharing schemes from three classes of linear codes.IEEE Trans. Inform. Theory, 52(1):206–212, 2006

    2006

  35. [35]

    X. Zhu, Y. Wu, and Q. Yue. New quaternary codes derived from posets of the disjoint union of two chains.IEEE Communications Letters, 24(1):20–24, 2019. 35

    2019