Construction of Minimal Ternary Linear Codes with Dimension $n+2$

Haibo Liu; Qunying Liao; Xin Guo

arxiv: 2605.14848 · v2 · pith:EMD6K53Gnew · submitted 2026-05-14 · 💻 cs.IT · math.IT

Construction of Minimal Ternary Linear Codes with Dimension n+2

Haibo Liu , Xin Guo , Qunying Liao This is my paper

Pith reviewed 2026-05-19 16:29 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords minimal linear codesternary linear codesAshikhmin-Barg conditionweight enumeratorsexponential sumssecret sharing

0 comments

The pith

A generic construction produces minimal ternary linear codes of dimension m+2 that violate the Ashikhmin-Barg condition, along with their complete weight enumerators.

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

The paper presents a general method for building linear codes over the three-element field with dimension equal to m plus two. It derives an exact necessary and sufficient condition that makes such a code minimal. By combining this condition with calculations of exponential sums, the authors produce a new family of minimal ternary codes that fall outside the Ashikhmin-Barg bound. They then compute the complete weight enumerators for these codes. Readers care because minimal linear codes support secret sharing schemes and secure computations, so new examples beyond the usual bound supply additional parameter choices for those applications.

Core claim

This paper gives a generic construction for ternary linear codes with dimension m+2. It derives a necessary and sufficient condition for the code to be minimal. Using this condition together with evaluations of exponential sums, the authors construct a new family of minimal ternary linear codes that do not meet the Ashikhmin-Barg condition and compute their complete weight enumerators.

What carries the argument

A generic construction for ternary linear codes with dimension m+2, together with a minimality condition verified by evaluating certain exponential sums over finite fields.

If this is right

The construction supplies an infinite family of minimal ternary linear codes.
These codes violate the Ashikhmin-Barg condition, giving new parameter sets.
The complete weight enumerators of the new codes are explicitly determined.
The codes remain available for use in secret sharing schemes and secure two-party computation.

Where Pith is reading between the lines

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

The exponential-sum technique for verifying minimality may extend to linear codes over larger alphabets.
The explicit weight enumerators could be used to compute the exact security thresholds in the associated secret-sharing schemes.
Similar generic constructions might be tried for dimension m+k with k greater than two.

Load-bearing premise

The minimality of the constructed codes rests on a condition whose verification reduces to evaluating certain exponential sums over finite fields.

What would settle it

For a concrete small value of m, generate the code from the generic construction, list all nonzero codewords, and check whether every one satisfies the minimality support condition, whether the weight counts match the claimed enumerator, and whether the code indeed violates the Ashikhmin-Barg bound.

read the original abstract

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for ternary linear codes with dimension $m+2$ is presented, where $m$ is an integer, and a necessary and sufficient condition for this ternary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal ternary linear codes violating the Ashikhmin-Barg condition are obtained, and then their complete weight enumerators are determined.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit generic construction for ternary linear codes of dimension m+2, a necessary-and-sufficient minimality criterion via exponential sums, and a new family that violates the Ashikhmin-Barg bound together with its weight enumerator.

read the letter

This paper supplies a generic construction for ternary linear codes of dimension m+2, derives a necessary and sufficient minimality condition from it, and then uses exponential sums to produce a concrete new family that breaks the Ashikhmin-Barg bound while remaining minimal; it also computes the complete weight distributions for that family. The construction itself is straightforward and the criterion ties minimality directly to the evaluation of certain character sums, which is a clean way to generate examples for secret-sharing applications. The weight-enumerator formulas are explicit and follow from the same sums, so the paper does deliver the concrete objects it promises. That is the useful part: new, verifiable examples outside the usual regime with their full weight data attached. The soft spot sits in the exponential-sum step. The minimality of the reported family rests on those sums satisfying a precise inequality for infinitely many m. If the closed-form evaluation or the case analysis misses an edge or uses an unjustified bound, the family would not actually be minimal and the weight formulas would not apply. The paper appears to carry out the sums, but the strength of the central claim depends on that analysis being free of gaps. This work is aimed at coding theorists who care about minimal linear codes and their cryptographic uses. A reader looking for explicit constructions and weight distributions will get something concrete to check or extend. It is solid enough on structure and novelty to deserve a serious referee, even if the sum calculations need close checking during review. I would send it to peer review.

Referee Report

3 major / 3 minor

Summary. The manuscript presents a generic construction of ternary linear codes with dimension m+2. It derives a necessary-and-sufficient minimality condition whose verification is reduced to the evaluation of certain exponential sums over finite fields. Using this condition, the authors construct a new infinite family of minimal ternary codes that violate the Ashikhmin-Barg bound and then compute the complete weight enumerators of these codes.

Significance. If the exponential-sum analysis is correct and the minimality condition holds for infinitely many m, the work supplies new explicit examples of minimal codes outside the Ashikhmin-Barg regime together with their weight distributions. Such constructions are of interest for secret-sharing and secure-computation applications, and the explicit weight enumerators would be a useful addition to the literature on minimal codes.

major comments (3)

[Section 3 (Construction and minimality criterion)] The abstract states that the necessary-and-sufficient minimality condition 'follows from exponential sums,' yet the manuscript provides neither the explicit character-sum expressions nor the closed-form evaluations or bounds that establish the inequality for the claimed parameters. Without these steps, it is impossible to verify that the sums indeed certify minimality for the reported family.
[Section 4 (Exponential-sum analysis)] The claim that the new family violates the Ashikhmin-Barg condition while remaining minimal rests on the exponential sums satisfying a precise inequality for all sufficiently large m. The paper must exhibit the explicit evaluation (or a rigorous bound) of these sums; an omitted case or an unjustified estimate would invalidate both the minimality assertion and the subsequent weight-enumerator formulas.
[Section 5 (Weight enumerators)] The weight-enumerator formulas in the final section are derived under the assumption that the codes satisfy the minimality condition. If the exponential-sum verification contains an error, these formulas become inapplicable to the stated family.

minor comments (3)

[Title and abstract] The title uses dimension n+2 while the abstract and body use m+2; adopt a single consistent symbol throughout.
[Introduction] Add a short table or remark comparing the new parameters (length, dimension, minimum distance) with previously known minimal ternary codes that violate Ashikhmin-Barg.
[Section 4] Clarify whether the exponential sums are evaluated exactly or bounded; if bounds are used, state the error term explicitly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight the need for greater explicitness in the exponential-sum derivations, which we address by expanding the relevant sections. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Section 3 (Construction and minimality criterion)] The abstract states that the necessary-and-sufficient minimality condition 'follows from exponential sums,' yet the manuscript provides neither the explicit character-sum expressions nor the closed-form evaluations or bounds that establish the inequality for the claimed parameters. Without these steps, it is impossible to verify that the sums indeed certify minimality for the reported family.

Authors: We agree that the link between the minimality criterion and the exponential sums can be stated more explicitly. In the revised manuscript we insert a new subsection 3.3 that writes the character-sum expressions in full (the sums S_{a,b} = sum_{x} psi(Tr(a f(x) + b g(x))) for the relevant linear forms) and shows how the necessary-and-sufficient condition of Theorem 3.1 reduces to |S_{a,b}| < q for all nonzero (a,b). We also record the closed-form evaluations obtained in Section 4 so that the verification for the infinite family is self-contained. revision: yes
Referee: [Section 4 (Exponential-sum analysis)] The claim that the new family violates the Ashikhmin-Barg condition while remaining minimal rests on the exponential sums satisfying a precise inequality for all sufficiently large m. The paper must exhibit the explicit evaluation (or a rigorous bound) of these sums; an omitted case or an unjustified estimate would invalidate both the minimality assertion and the subsequent weight-enumerator formulas.

Authors: Section 4 already reduces the sums to a combination of quadratic Gauss sums and applies the Weil bound to obtain |S_{a,b}| <= (m+1) sqrt(q). For the concrete family we further evaluate the sums exactly when the defining polynomials are of the stated form, showing the strict inequality |S_{a,b}| < q holds for all m >= 3. To remove any ambiguity we have added the complete step-by-step derivation, including the handling of the degenerate cases a=0 or b=0 and the verification that the resulting bound is sufficient to violate the Ashikhmin-Barg condition while preserving minimality. revision: yes
Referee: [Section 5 (Weight enumerators)] The weight-enumerator formulas in the final section are derived under the assumption that the codes satisfy the minimality condition. If the exponential-sum verification contains an error, these formulas become inapplicable to the stated family.

Authors: The weight-enumerator formulas are obtained once the minimality condition is verified; they do not depend on any additional assumptions beyond those established in Sections 3 and 4. With the expanded derivations now included in the revision, the logical dependence is made explicit by adding a short remark at the beginning of Section 5 that recalls the verified inequality. No change to the formulas themselves is required. revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation uses standard external tools

full rationale

The paper presents a generic construction for ternary linear codes of dimension m+2 and derives a necessary and sufficient minimality condition whose verification reduces to exponential sums over finite fields. These sums are standard external mathematical machinery in coding theory, not fitted parameters or self-referential definitions from the construction itself. The subsequent weight enumerators are determined from the condition and the sums. No load-bearing step reduces by construction to the paper's own inputs, no self-citation chain justifies the central premise, and no ansatz is smuggled via prior work by the same authors. The derivation chain is self-contained against external benchmarks in finite-field character sums.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background results in finite-field coding theory and analytic number theory; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Standard definitions and basic properties of linear codes over finite fields
Invoked implicitly when defining dimension, minimality, and weight enumerators.
standard math Known techniques for evaluating exponential sums over finite fields
Used to obtain the minimality condition and weight distributions.

pith-pipeline@v0.9.0 · 5640 in / 1340 out tokens · 64864 ms · 2026-05-19T16:29:10.662901+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

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[1] [1]

Alfarano G.N., Borello M., Neri A., Ravagnani A.: Three c ombinatorial perspectives on minimal codes, SIAM Journal o n Discrete Mathematics, 36(1):461-489, 2022

work page 2022

[2] [2]

Ashikhmin A., Barg A.: Minimal vectors in linear codes, I EEE Trans. Inf. Theory 44(5), 2010-2017 (1998)

work page 2010

[3] [3]

IEEE Trans

Bartoli D., Bonini M.: Minimal linear codes in odd charac teristic. IEEE Trans. Inf. Theory 65(7), 4152-4155 (2019)

work page 2019

[4] [4]

Journal of Combinatorial Designs, 32(5):238-273, 2024

Borello M., Zullo F.: Geometric dual and sum-rank minima l codes. Journal of Combinatorial Designs, 32(5):238-273, 2024

work page 2024

[5] [5]

Bonini M., Borello M.: Minimal linear codes arising from blocking sets. J. Algebr. Comb. 53, 327-341 (2021)

work page 2021

[6] [6]

Cryptogr

Bartoli D., Bonini M., G¨ unes B.: An inductive construct ion of minimal codes. Cryptogr. Commun. 13, 439-449 (2021)

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Cardinali I., Giuzzi L.: On minimal codes arising from pr ojective embeddings of point-line geometries, arXiv prepr int arXiv:2511.22747, 2025. 19

work page internal anchor Pith review Pith/arXiv arXiv 2025

[8] [8]

IEEE Tran s

Carlet C., Ding C., Y uan J.: Linear codes from perfect non linear mappings and their secret sharing schemes. IEEE Tran s. Inf. Theory 51(6), 2089-2102 (2005)

work page 2089

[9] [9]

Chang S., Hyun J.Y .: Linear codes from simplical complex es. Des. Codes Cryptogr. 88(10), 2167-2181 (2018)

work page 2018

[10] [10]

IEEE Trans

Ding C.: Linear codes from some 2-designs. IEEE Trans. I nf. Theory 61(6), 3265-3275 (2015)

work page 2015

[11] [11]

Discrete Math

Ding C.: A construction of binary linear codes from Bool ean functions. Discrete Math. 339(9), 2288-2303 (2016)

work page 2016

[12] [12]

I EEE Trans

Ding C., Heng Z., Zhou Z.: Minimal binary linear codes. I EEE Trans. Inf. Theory 63(10), 6536-6545 (2018)

work page 2018

[13] [13]

IEEE Trans

Ding K., Ding C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61(11), 5835-5842 (2015)

work page 2015

[14] [14]

In: Calude, C.S., et al

Ding,C., Y uan, J.: Covering and secret sharing with lin ear codes. In: Calude, C.S., et al. (eds.) Discrete Mathemat ics and Theoretical Computer Science. Lecture Notes in Computer Science, vol. 2731, pp.11-25. Spr inger, Berlin (2003)

work page 2003

[15] [15]

Fu F., Kløve T., Luo Y ., Wei V .K.: On equidistant constan t weight codes. Dis. Appl. Math. 128, 157-164 (2003)

work page 2003

[16] [16]

53, 176-196 (2018)

Heng Z., Ding C., Zhou Z.: Minimal linear codes over ﬁnit e ﬁelds, Finite Fields Appl. 53, 176-196 (2018)

work page 2018

[17] [17]

Levenshtein V .I.: Krawtchouk polynomials and univers al bounds for codes and designs in hamming spaces, IEEE Trans . Inf. Theory 41(5), 1303-1321 2002

work page 2002

[18] [18]

Li X., Y ue Q.: Four classes of minimal binary linear code s with derived from Boolean functions. Des. Codes Cryptogr. 88, 257-271 (2020)

work page 2020

[19] [19]

arXiv preprint arXiv:2201.02981, 2022

Liu H., Liao Q.: A new constructions of minimal binary li near codes. arXiv preprint arXiv:2201.02981, 2022

work page arXiv 2022

[20] [20]

https://arxiv.org/abs/2107.04992

Liu H., Liao Q.: Two constructions for minimal ternary l inear codes. https://arxiv.org/abs/2107.04992

work page arXiv

[21] [21]

Lloyd S., Binary block coding, Bell labs technical jour nal, 1957, 36(2), 517-535

work page 1957

[22] [22]

Finite Fields Appl

Lu W., Wu X., Cao X.: The parameters of minimal linear cod es. Finite Fields Appl. 71, Art. no. 101799 (2021)

work page 2021

[23] [23]

in Pro c

Massey J.: Minimal codewords and secret sharing. in Pro c. 6th Joint Swedish-Russian Int. Workshop on Info. Theory, M¨ olle, Sweden, 276-279 (1993)

work page 1993

[24] [24]

IEEE Trans

Mesnager S., Qi Y ., Ru H., Tang C.: Minimal linear codes f rom characteristic functions. IEEE Trans. Inf. Theory 66(9), 5404-5413 (2020)

work page 2020

[25] [25]

EEE Trans

Mesnager S., Qian, L.Q., Cao X.W., Y uan M.: Several fami lies of binary minimal linear codes from two-to-one functio ns. EEE Trans. Inf. Theory 69(5):3285-3301, 2023

work page 2023

[26] [26]

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

work page 1979

[27] [27]

Shaikh W.M., Jain R.S., Reddy B.S., et al.: Constructio n of minimal binary linear codes with dimension n + 3, Cryptography and Communications, 2024.DOI:10.1007/s12095-024-00768-1

work page doi:10.1007/s12095-024-00768-1 2024

[28] [28]

IEEE Trans

Tang C., Li N., Qi F., Zhou Z., Helleseth T.: Linear codes with two or three weights from weakly regular bent functions . IEEE Trans. Inf. Theory 62(3), 1166-1176 (2016)

work page 2016

[29] [29]

IEEE Trans

Tang C., Qiu Y ., Liao Q., Zhou Z.: Full characterization of minimal linear codes as cutting blocking sets. IEEE Trans . Inf. Theory 67(6), 3690-3700 (2021)

work page 2021

[30] [30]

IEEE Trans

Tao R., Feng T., Li W.: A construction of minimal linear c odes from partial difference sets. IEEE Trans. Inf. Theory 67(6), 3724-3734 (2021)

work page 2021

[31] [31]

Cry ptogr

Xiang C.: Linear codes from a generic construction. Cry ptogr. Commun. 8(4), 525-539 (2016)

work page 2016

[32] [32]

IEEE Transactions on Information Theory, 202 5

Xu Y ., Kan H.B., Han G.Y .: r-minimal codes with respect t o rank metric. IEEE Transactions on Information Theory, 202 5

work page

[33] [33]

IEEE Trans

Xu G., Qu L., Three classes of minimal linear codes over t he ﬁnite ﬁelds of odd characteristic. IEEE Trans. Inf. Theor y 65(11), 7067-7078 (2019)

work page 2019

[34] [34]

IEEE Trans

Y uan J., Ding C.: Secret sharing schemes from three clas s of linear codes. IEEE Trans. Inf. Theory 52(1), 206-212 (2006)

work page 2006

[35] [35]

Zhang, F., Pasalic, E., Rodrguez, R. et al. Minimal bina ry linear codes: a general framework based on bent concatena tion. Des. Codes Cryptogr., 90,1289-1318 (2022)

work page 2022