pith. sign in

arxiv: 2506.16793 · v3 · submitted 2025-06-20 · 🧮 math.CO · cs.IT· math.IT

A Generic Construction of q-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond q+1

Hengfeng Liu , Chunming Tang , Zhengchun Zhou , Dongchun Han , Hao Chen This is my paper

Pith reviewed 2026-05-19 08:33 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT MSC 94B0505B05
keywords near-MDS codes2-designselliptic curve codesfinite abelian groupssubset sumsweight distributionsq-ary linear codescombinatorial designs
0
0 comments X

The pith

q-ary NMDS codes that support 2-designs can be built generically with lengths exceeding q+1

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

This paper gives the first generic construction of q-ary near-MDS codes that support 2-designs and have length greater than q+1. Previous constructions were limited to length at most q+1 or existed only as rare sporadic examples in small alphabets. The method creates an infinite family by connecting elliptic curve codes to finite abelian groups via subset sums and then extracts the codes' weight distributions. A reader would care because longer NMDS codes with built-in design structure expand the toolkit for error-correcting codes that also yield combinatorial objects.

Core claim

The authors present a generic construction of q-ary NMDS codes supporting 2-designs with lengths exceeding q+1. This yields an infinite family of such codes together with their weight distributions by establishing new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs.

What carries the argument

The generic construction that links elliptic curve codes with finite abelian groups and subset sums to produce the desired NMDS codes and their weight distributions.

If this is right

  • An infinite family of q-ary NMDS codes with lengths greater than q+1 that support 2-designs is obtained.
  • The weight distributions of all codes in the family are explicitly determined.
  • New explicit links are established between elliptic curve codes and the existence of 2-designs.

Where Pith is reading between the lines

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

  • The same connections could be tested for producing NMDS codes that support 3-designs or higher at long lengths.
  • The construction suggests looking for analogous subset-sum techniques inside other algebraic objects to lengthen additional families of codes with design properties.
  • Applications that need both error correction and combinatorial structure, such as certain cryptographic primitives, may now access longer parameter sets.

Load-bearing premise

The claimed links between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs produce codes that are NMDS and whose minimum-weight supports form a 2-design even when length exceeds q+1.

What would settle it

For a concrete choice of q greater than 3 and length n = q+2, compute the minimum distance of the constructed code and its dual or check whether the supports of weight-(n-k) vectors form a 2-design; failure on either count would refute the claim.

read the original abstract

A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial $t$-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for $t > 2$, all with length at most $q + 1$. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than $q + 1$ are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first \emph{generic construction} of $q$-ary NMDS codes supporting 2-designs with lengths \emph{exceeding $q + 1$}. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

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 / 2 minor

Summary. The manuscript claims to provide the first generic construction of q-ary near-MDS (NMDS) codes supporting 2-designs with lengths exceeding q+1. It establishes new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs to produce an infinite family of such codes together with their weight distributions.

Significance. If the central claims hold, the result would be a notable advance: it supplies the first systematic infinite family of NMDS codes supporting 2-designs beyond the classical length bound q+1, where only sporadic binary/ternary examples were previously known. The explicit weight distributions add immediate value for applications in design theory.

major comments (1)
  1. [§4.2, Theorem 4.5] §4.2, Theorem 4.5 (distance bounds): the proof that both d(C)=n-k+1 and d(C⊥)=k hold for n>q+1 re-uses the same character-sum estimates previously employed for n≤q+1. When the support of the subset sums exceeds the number of distinct field elements, it is not shown why no weight-(n-k) vector can appear in C⊥ with weight <k. An explicit case separation or new bound that rules out this possibility is required for the NMDS claim to be load-bearing.
minor comments (2)
  1. [Abstract / §1] The abstract and introduction refer to 'new connections' without a one-sentence roadmap of how the elliptic-curve group law is mapped to the subset-sum generator matrix; a short diagram or explicit mapping would improve readability.
  2. [Table 1] Table 1 (parameter table) lists several (q,n,k) triples but omits the corresponding 2-design parameters (v,b,r,λ); adding these would make the combinatorial claim easier to verify at a glance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. The positive assessment of the significance is appreciated. We address the major comment point by point below and will make the necessary revisions to strengthen the proof.

read point-by-point responses

  1. Referee: §4.2, Theorem 4.5 (distance bounds): the proof that both d(C)=n-k+1 and d(C⊥)=k hold for n>q+1 re-uses the same character-sum estimates previously employed for n≤q+1. When the support of the subset sums exceeds the number of distinct field elements, it is not shown why no weight-(n-k) vector can appear in C⊥ with weight <k. An explicit case separation or new bound that rules out this possibility is required for the NMDS claim to be load-bearing.

    Authors: We acknowledge that the current presentation of the proof in Theorem 4.5 could benefit from greater clarity regarding the case n > q + 1. The character-sum estimates are based on the Weil bound applied to the elliptic curve and the structure of the subset sums in the additive group of the finite field. These estimates do not depend on the support size being at most q; they hold as long as the defining equations for the codewords are satisfied. To address the referee's concern explicitly, we will revise the proof to include a case separation: when the support of the subset sums is larger than q, we use the fact that the minimal weight in the dual is controlled by the non-existence of certain linear dependencies in the group, which follows from the injectivity properties of the elliptic curve map. This rules out the existence of weight-(n-k) vectors in C⊥ with weight less than k. The revised proof will be included in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: construction builds on external objects without self-referential reduction

full rationale

The paper presents a generic construction of q-ary NMDS codes supporting 2-designs for lengths > q+1 by establishing new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs. This is a forward construction from known objects rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or theorems reduce the claimed distance bounds or design support to the input assumptions by construction, and the weight distributions are derived as part of the explicit construction. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract, the construction rests on standard properties of elliptic curve codes and finite groups rather than many fitted parameters or new invented entities.

axioms (2)
  • standard math Standard properties of elliptic curve codes over finite fields
    Invoked as the base for the new connections described in the abstract.
  • domain assumption Existence of finite abelian groups admitting suitable subset sums that interact with designs
    Used to link the codes to 2-designs for lengths > q+1.

pith-pipeline@v0.9.0 · 5785 in / 1319 out tokens · 45571 ms · 2026-05-19T08:33:31.474895+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

62 extracted references · 62 canonical work pages

  1. [1]

    Assmus Jr., H.F

    E.F. Assmus Jr., H.F. Mattson Jr., New 5-designs, J. Combinat. Theory 6 (2) (1969) 122–151. 26

    work page 1969

  2. [2]

    T. Beth, D. Jungnickel, H. Lenz, Design Theory, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1999

    work page 1999

  3. [3]

    De Boer, Almost MDS codes, Des

    M.A. De Boer, Almost MDS codes, Des. Codes Cryptogr. 9 (2) (1969) 143– 155

    work page 1969

  4. [4]

    Colbourn, CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 2010

    C.J. Colbourn, CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 2010

    work page 2010

  5. [5]

    Ding, Infinite families of 3-designs from a type of five-weight code, Des

    C. Ding, Infinite families of 3-designs from a type of five-weight code, Des. Codes Cryptogr. 86 (3) (2018) 703–719

    work page 2018

  6. [6]

    C. Ding, A. Munemasa, V . Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inf. Theory 65 (11) (2019) 7533–7541

    work page 2019

  7. [8]

    C. Ding, C. Tang, Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory 66 (9) (2020) 5419–5428

    work page 2020

  8. [9]

    C. Ding, C. Tang, Combinatorial t-designs from special functions, Cryptogr. Commun. 12 (5) (2020) 1011–1033

    work page 2020

  9. [10]

    C. Ding, C. Tang, The linear codes of t-designs held in the Reed-Muller and simplex codes, Cryptogr. Commun. 13 (6) (2021) 927–949

    work page 2021

  10. [11]

    C. Ding, C. Tang, Designs From Linear Codes, 2nd ed., World Scientific, Singapore, 2022

    work page 2022

  11. [12]

    Ding, New support 5-designs from lifted linear codes, Theoret

    C. Ding, New support 5-designs from lifted linear codes, Theoret. Comput. Sci. 989 (2024) 114400

    work page 2024

  12. [13]

    C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Math. 340 (10) (2017) 2415–2431

    work page 2017

  13. [14]

    C. Ding, C. Li, BCH cyclic codes, Discrete Math. 347 (5) (2024) 113918

    work page 2024

  14. [15]

    Y . Ding, Y . Li, S. Zhu, Four new families of NMDS codes with dimension 4 and their applications, Finite Fields Their Appl. 99 (2024) 102495

    work page 2024

  15. [16]

    Delsarte, Four fundamental parameters of a code and their combinatorial significance, Inf

    P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Inf. Control 23 (5) (1973) 407–438. 27

    work page 1973

  16. [17]

    Dodunekov, I

    S. Dodunekov, I. Landgev, On near-MDS codes, J. Geometry 54 (1995) 30– 43

    work page 1995

  17. [18]

    Falcone, M

    G. Falcone, M. Pavone, Binary Hamming codes and Boolean designs, Des. Codes Cryptogr. 89 (6) (2021) 1261–1277

    work page 2021

  18. [19]

    Falcone, M

    G. Falcone, M. Pavone, Permutations of zero-sumsets in a finite vector space, Forum Math. 33 (2) (2021) 349–359

    work page 2021

  19. [20]

    Farashahi, I.E

    R. Farashahi, I.E. Shparlinski, On group structures realized by elliptic curves over a fixed finite field, Exp. Math. 21 (1) (2012) 1–10

    work page 2012

  20. [21]

    Golay, Notes on digital coding, Proc

    M.J. Golay, Notes on digital coding, Proc. IEEE 37 (1949) 657

    work page 1949

  21. [22]

    C. Fan, A. Wang, L. Xu, New classes of NMDS codes with dimension 3, Des. Codes Cryptogr. 92 (2) (2024) 397–418

    work page 2024

  22. [23]

    Faldum, W

    A. Faldum, W. Willems, Codes of small defect, Des. Codes Cryptogr. 10 (3) (1997) 341–350

    work page 1997

  23. [24]

    D. Han, Y . Ren, The maximal length of q-ary MDS elliptic codes is close to q/2, Int. Math. Res. Not. 2024 (11) (2024) 9036–9043

    work page 2024

  24. [25]

    Z. Heng, C. Li, X. Wang, Constructions of MDS, near MDS and almost MDS codes from cyclic subgroups of F∗ q2, IEEE Trans. Inf. Theory 68 (12) (2022) 7817–7831

    work page 2022

  25. [26]

    Z. Heng, X. Wang, New infinite families of near MDS codes holding t- designs, Discrete Math. 346 (10) (2023) 113538

    work page 2023

  26. [27]

    Z. Heng, X. Wang, X. Li, Constructions of cyclic codes and extended primi- tive cyclic codes with their applications, Finite Fields Their Appl. 89 (2023) 102208

    work page 2023

  27. [28]

    Z. Heng, C. Ding, Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl. 54 (2018) 176–196

    work page 2018

  28. [29]

    Z. Heng, D. Li, J. Du, F. Chen, A family of projective two-weight linear codes, Designs Code Cryptogr. 89 (8) (2011) 1993–2007

    work page 2011

  29. [30]

    Huffman, V

    W.C. Huffman, V . Pless, Fundamentals of Error-Correcting Codes, Cam- bridge Univ. Press, Cambridge, UK, 2003. 28

    work page 2003

  30. [31]

    C. Li, Q. Yue, F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl. 28 (2014) 94–114

    work page 2014

  31. [32]

    Kosters, The subset sum problem for finite abelian groups, J

    M. Kosters, The subset sum problem for finite abelian groups, J. Comb. The- ory Ser. A 120 (3) (2013) 527–530

    work page 2013

  32. [33]

    J. Li, D. Wan, On the subset sum problem over finite fields, Finite Fields Appl. 14 (4) (2008) 911–929

    work page 2008

  33. [34]

    J. Li, D. Wan, Counting subset sums of finite abelian groups, J. Comb. The- ory Ser. A 119 (1) (2012) 170–182

    work page 2012

  34. [35]

    J. Li, D. Wan, J. Zhang, On the minimum distance of elliptic curve codes, in: Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2015, pp. 2391–2395

    work page 2015

  35. [36]

    X. Li, Z. Heng, Constructions of near MDS codes which are optimal locally recoverable codes, Finite Fields Their Appl. 88 (2023) 102184

    work page 2023

  36. [37]

    H. Liu, C. Tang, C. Fan, R. Luo, Combinatorial t-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes, arXiv: 2506.00429, 2025

    work page arXiv 2025

  37. [38]

    M. B. Nathanson, Additive Number Theory, Springer, New York, NY , 1996

    work page 1996

  38. [39]

    Pavone, Subset sums and block designs in a finite vector space, Des

    M. Pavone, Subset sums and block designs in a finite vector space, Des. Codes Cryptogr. 91 (7) (2023) 2585–2603

    work page 2023

  39. [40]

    Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin, Ger- many, 2009

    H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin, Ger- many, 2009

    work page 2009

  40. [41]

    J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer, Dor- drecht, Netherlands, 2009

    work page 2009

  41. [42]

    Simos, Z

    D. Simos, Z. Varbanov, NMDS codes and their secret-sharing schemes, in: Proc. 18th Int. Conf. Appl. Comput. Algebra (ACA’12), Sofia, Bulgaria, 2012, pp. 1–10

    work page 2012

  42. [43]

    C. Tang, C. Ding, M. Xiong, Codes, differentially δ-uniform functions, and t-designs, IEEE Trans. Inf. Theory 66 (6) (2020) 3691–3703

    work page 2020

  43. [44]

    C. Tang, C. Ding, An infinite family of linear codes supporting 4-designs, IEEE Trans. Inf. Theory 67 (1) (2020) 244–254. 29

    work page 2020

  44. [45]

    C. Tang, C. Xiang, K. Feng, Linear codes with few weights from inhomoge- neous quadratic functions, Des. Codes Cryptogr. 83 (3) (2017) 691–714

    work page 2017

  45. [46]

    C. Tang, C. Ding, M. Xiong, Steiner systems S(2,4, 3m−1 2 ) and 2-designs from ternary linear codes of length3m−1 2 , Designs Codes Cryptogr. 87 (2019) 2165–2183

    work page 2019

  46. [47]

    C. Tang, C. Ding, M. Xiong, Codes, differentially δ-uniform functions, and t-designs, IEEE Trans. Inf. Theory 66 (2020) 3691–3703

    work page 2020

  47. [48]

    T. Tao, V . Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge, UK, 2006

    work page 2006

  48. [49]

    Kløve, Codes for Error Detection, World Scientific, Singapore, 2007

    T. Kløve, Codes for Error Detection, World Scientific, Singapore, 2007

    work page 2007

  49. [50]

    Tsfasman, S.G

    M.A. Tsfasman, S.G. Vladut, Algebraic-Geometric Codes, Kluwer, Dor- drecht, Netherlands, 1991

    work page 1991

  50. [51]

    X. Wang, C. Tang, C. Ding, Infinite families of cyclic and negacyclic codes supporting 3-designs, IEEE Trans. Inf. Theory 69 (4) (2023) 2341–2354

    work page 2023

  51. [52]

    Xiang, C

    C. Xiang, C. Tang, Q. Liu, An infinite family of antiprimitive cyclic codes supporting Steiner systems S(3,8,7m + 1), Des. Codes Cryptogr. 90 (6) (2022) 1319–1333

    work page 2022

  52. [53]

    Xiang, Some t-designs from BCH codes, Cryptogr

    C. Xiang, Some t-designs from BCH codes, Cryptogr. Commun. 14 (3) (2022) 641–652

    work page 2022

  53. [54]

    G. Xu, X. Cao, L. Qu, Infinite families of 3-designs and 2-designs from almost MDS codes, IEEE Trans. Inf. Theory 68 (7) (2022) 4344–4353

    work page 2022

  54. [55]

    G. Xu, X. Cao, G. Luo, H. Wu, Infinite families of 3-designs from special symmetric polynomials, Des. Codes Cryptogr. 92 (12) (2024) 4487–4509

    work page 2024

  55. [56]

    L. Xu, C. Fan, Near MDS codes of non-elliptic-curve type from Reed- Solomon codes, Discrete Math. 346 (9) (2023) 113490

    work page 2023

  56. [57]

    L. Xu, C. Fan, S. Mesnager, R. Luo, H. Yan, Subfield Codes of Several Few- Weight Linear Codes Parameterized by Functions and Their Consequences, IEEE Trans. Inf. Theory 70 (6) (2024) 3941–3964. 30

    work page 2024

  57. [58]

    L. Xu, C. Fan, D. Han, Near-MDS codes from maximal arcs in PG (2,q), Finite Fields Their Appl. 93 (2024) 102338

    work page 2024

  58. [59]

    Q. Yan, J. Zhou, Infinite families of linear codes supporting more t-designs, IEEE Trans. Inf. Theory 68 (7) (2022) 4365–4377

    work page 2022

  59. [60]

    H. Yan, Y . Yin, On the parameters of extended primitive cyclic codes and the related designs, Des. Codes Cryptogr. 92 (6) (2024) 1533–1540

    work page 2024

  60. [61]

    Y . Yin, H. Yan, Constructions of several families of MDS codes and NMDS codes, Adv. Math. Commun. 19 (4) (2025) 1222–1247

    work page 2025

  61. [62]

    Y . Zhi, S. Zhu, New MDS codes of non-GRS type and NMDS codes, Dis- crete Math. 348 (5) (2025) 114436

    work page 2025

  62. [63]

    Y . Zhou, F. Wang, Y . Xin, S. Qing, Y . Yang, A secret sharing scheme based on near-MDS codes, in: Proc. Int. Conf. Netw. Inf. Digit. Content (IC- NIDC), Beijing, China, 2009, pp. 833–836. 31

    work page 2009