Minimal linear codes from characteristic functions

Chunming Tang; Hongming Ru; Sihem Mesnager; Yanfeng Qi

REVIEW 3 major objections 5 minor 39 references

Minimal linear codes from characteristic functions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Characteristic functions turn minimality testing and construction of linear codes into Walsh-transform computations, and yield infinite families of minimal codes with weight ratio at most $(p-1)/p$.

desk verdict Section 3 has salvageable constructions, but the advertised Walsh characterization is wrong as written: Theorem 4.2's sign error certifies a non-minimal code, so the paper needs major revision before it can be trusted. read the letter →

arxiv 1908.01650 v2 pith:ZFPYMYA7 submitted 2019-08-05 cs.IT math.IT

Sihem Mesnager , Yanfeng Qi , Hongming Ru , Chunming Tang This is my paper · ORCID

classification cs.ITmath.IT MSC 94B0511T7194B65

keywords minimallinearcodescharacteristicfunctionsWalshtransformweightdistributiondefiningsetmethodsubspaceconstructionsKrawchoukpolynomialsfinitefields

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Minimal linear codes—codes in which no nonzero codeword's support contains another codeword's support—are sought after for secret-sharing and secure two-party computation, but they are hard to certify. The paper claims that characteristic functions of subsets of a finite field turn the search into Walsh-transform control: minimality of a defining-set code is equivalent to an explicit inequality in Walsh sums, with a particularly simple form in the binary case. Complementing a defining set changes the Walsh spectrum by a fixed affine shift, so every code comes with a companion code whose weight distribution is read off from the original spectrum. The same functions, applied to unions of carefully separated subspaces, yield infinite families of minimal codes whose lightest nonzero weight is at most $(p-1)/p$ times the heaviest, and a concrete family from all vectors of Hamming weight one or two is minimal for every prime $p$.

What carries the argument

The load-bearing object is the characteristic function $f_D(x)=1$ if $x\in D$, $0$ otherwise, and its Walsh transform $\hat f_D(w)=\sum_{x\in\mathbb{F}_q}\zeta_p^{f_D(x)-\operatorname{Tr}(wx)}$. Two identities carry the argument: complementing $D$ affinely shifts the Walsh spectrum (Lemma 3.1), and the Walsh transform of a disjoint union is the sum of the constituent transforms (Lemma 3.2). For subspace-defined sets $D=\cup_i E_i\setminus\{0\}$, formula (9) expresses every Walsh value in terms of the subspace dimensions and orthogonality pattern, and Proposition 3.7 classifies which subspace arrangements can satisfy the required disjointness assumptions. The minimality test itself is the criterion of Theorem 1.2: a linear code is minimal if and only if $\sum_{c\in\mathbb{F}_p^*}\operatorname{wt}(a+cb)\neq(p-1)\operatorname{wt}(a)-\operatorname{wt}(b)$ for every pair of linearly independent codewords $a,b$. Krawchouk polynomials supply the Walsh values for Hamming-ball defining sets such as the weight-one-or-two family.

What would settle it

Enumerate every subset $D\subseteq\mathbb{F}_2^m\setminus\{0\}$ for a small $m$ (say $m=5$ or $6$), compute $\hat f_D$ for all nonzero arguments, and compare the binary characterization $\hat f_D(\beta_1+\beta_2)-\hat f_D(\beta_1)-\hat f_D(\beta_2)\neq 2|D|$ with a direct brute-force verdict on whether $C_D$ is minimal; a single mismatch would disprove Theorem 4.2.

Watch

Extended reading notes

Core claim

The paper's central claim, on its own terms, is that the Walsh transform of the characteristic function $f_D$ of a defining set $D$ completely governs whether the code $C_D=\{(\operatorname{Tr}(\beta x))_{x\in D}:\beta\in\mathbb{F}_q\}$ is minimal. For general $p$, $C_D$ is minimal exactly when condition (11) holds—the printed inequality involving $\sum_{y\in\mathbb{F}_p^*}(\sum_{c\in\mathbb{F}_p^*}\hat f_D(y\beta_1+yc\beta_2)+\hat f_D(y\beta_2)-(p-1)\hat f_D(y\beta_1))$ and $(\zeta_p-1)(p-1)|D|$—for every pair of linearly independent $\beta_1,\beta_2$. In the binary case this collapses to the condition $\hat f_D(\beta_1+\beta_2)-\hat f_D(\beta_1)-\hat f_D(\beta_2)\neq 2|D|$ for all distinct nonzero $\beta_1,\beta_2$, with the sufficient bound $|\hat f_D(\beta)|<2|D|/3$. The paper also claims that characteristic functions of unions of pairwise separated subspaces produce many minimal codes, including binary and odd-characteristic families reaching $w_{\min}/w_{\max}\le(p-1)/p$, and that the set of vectors of Hamming weight one or two always gives a minimal defining-set code.

Load-bearing premise

The subspace-built weight formulas and minimality theorems rest on condition (8): every pair of chosen subspaces must intersect only at zero, and the same must hold for their orthogonal complements; Proposition 3.7 shows this assumption allows only three structural configurations, so the paper's 'many minimal codes' should be read as many parameter choices within those three shapes.

Editorial extensions

If this is right

In the binary case, minimality of any defining-set code can be certified by checking one Walsh-transform inequality over all distinct nonzero pairs, giving a finite and explicit test.
Every binary defining set $D$ yields a companion code from its complement whose weight distribution is obtained by replacing $\hat f_D$ with $2-\hat f_D$; the odd-characteristic analogue follows from Lemma 3.1 and Corollary 3.4.
When $m$ is even, unions of $s$ half-size subspaces give minimal binary codes exactly when $s$ avoids the two forbidden values, and the forbidden values are the only obstruction in that family.
The defining set $D_{12}=\{\beta:\,1\le\operatorname{wt}(\beta)\le2\}$ gives a minimal code over every prime $p$, and for $m\ge6$ its weight ratio is at most $(p-1)/p$.
The same Walsh machinery applies to the original defining-set construction $C_D$ and to the $C_f$ construction, so the results unify two previously separate families.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: Theorem 4.2's iff condition suggests a direct algorithmic sieve—enumerate candidate defining sets in small fields and evaluate the two Walsh expressions—rather than checking all codeword cover relations as a direct minimality test would.
Editorial inference: Proposition 3.7's three-configuration classification implies the subspace route cannot produce arbitrary geometric variety; the reported 'many minimal codes' are many parameter families within one-subspace, two-complementary-subspace, or several-half-size-subspace layouts.
Editorial inference: The weight-one-or-two family points to a general pattern: defining sets formed from low-weight Hamming layers have Krawchouk-expressible Walsh transforms, so the same proof should test weight-three or weight-four layers for minimality.
Editorial inference: Combined with the LCD-code families cited in the paper, the two-subspace case $E_2=E_1^\perp$ turns any linear complementary dual code into a minimal code, offering a search path the paper mentions but does not develop.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the Section 3 constructions are real and probably salvageable, but the advertised Section 4 characterization is unsound as written. The stress-test example is correct: for m=2, D={(1,0),(0,1)}, the defining-set code is [2,2,1] with c_e1=(1,1), c_e2=(0,1), c_e1+e2=(1,0); c_e1 covers c_e2, so the code is not minimal, yet Theorem 4.2's inequality holds. The sign error is not cosmetic: specializing Theorem 4.1 to p=2 gives a different condition, and Theorem 4.2 can certify non-minimal codes. Since the abstract advertises the characterization as a main contribution, that claim fails.

What is genuinely useful: the complement-of-D trick and the subspace-union construction. Corollaries 3.3 and 3.4 give weight distributions for complements; Theorems 3.8–3.12 add families with explicit weight enumerators, including Examples 1–7. D12 for odd p appears new, and the w_min/w_max ratio bounds are computed carefully. The paper also points back to Ding et al. and Xu–Qu rather than overselling novelty.

Soft spots, in order of severity:

1. Theorem 4.2's condition is wrong. It is not the p=2 case of (11). The counterexample is small and unambiguous. Corollary 4.3 inherits the error. Any revision needs to fix signs and re-run the D12 minimality proof, which currently cites Theorem 4.1/4.2.

2. The proof of Theorem 3.5 checks \hat f_{\bar D} ± \hat f_D instead of the same function. Theorem 2.2 requires both conditions for one function. That proof needs repair; the weight tables may still be right, but the minimality conclusion lacks support as written.

3. Proposition 3.7 shows condition (8) is restrictive: only single subspace, two complementary subspaces, or several half-size subspaces with m even. So “many minimal codes” overstates the range. It does not kill the families, but the reader should know.

4. Minor: the Section 4 result is largely a translation of Heng–Ding–Zhou's criterion into Walsh language; the novel part is the concrete D12 class.

Who gets value: researchers working on defining-set constructions and few-weight minimal codes. The examples and weight tables are checkable and potentially useful. But the current version should not be accepted: the central characterization is false and the Section 3 proof has a gap. I would send it to a competent referee only if the authors are willing to fix Section 4 or drop it; otherwise desk reject.

Referee Report

3 major / 5 minor

Summary. The paper studies minimal linear codes built from characteristic functions of subsets of F_q, both in the construction C_f = {(u f(x) - Tr(vx))_{x in F_q^*}} and in the defining-set construction C_D = {(Tr(beta x))_{x in D}}. It reports weight distributions, constructs families of binary and odd-characteristic minimal codes from subspaces, and claims a Walsh-transform characterization of minimality for defining-set codes. The main advertised contribution is the binary characterization in Theorem 4.2 and the associated sufficient condition. The paper also gives several explicit examples and weight enumerators.

Significance. If correct, the paper would contribute a useful Walsh-domain criterion for minimality of defining-set codes, several new infinite families with w_min/w_max below or equal to (p-1)/p, and explicit weight distributions. The paper is notable for its concrete examples, detailed weight enumerators, and the systematic use of Walsh transforms. The subspace-based constructions in Section III.B appear to be a substantive contribution. However, the central Section 4 characterization is false as stated, and the proof of Theorem 3.5 has a gap in its application of Theorem 2.2; these issues affect claims advertised in the abstract and cannot be ignored.

major comments (3)

[Section 4, Theorem 4.2] Theorem 4.2 is false as stated. Take D={e_1, e_1+e_2} subset F_2^2. One computes hat f_D(e_1)=4, hat f_D(e_2)=0, hat f_D(e_1+e_2)=0, and |D|=2. For beta_1=e_1 and beta_2=e_2, the printed inequality hat f_D(beta_1+beta_2)-hat f_D(beta_1)-hat f_D(beta_2) != 2|D| reads -4 != 4, so the theorem predicts minimality. But the code C_D has codewords c_{e_1}=(1,1), c_{e_2}=(0,1), and c_{e_1+e_2}=(1,0); c_{e_1} covers c_{e_2}, so C_D is not minimal. The correct binary specialization of Theorem 4.1 is hat f_D(beta_1)-hat f_D(beta_1+beta_2)-hat f_D(beta_2) != 2|D|, which flags this example. Consequently Corollary 4.3 and the abstract's claim of a defining-set characterization are unsupported as stated.
[Section 3.A, Theorem 3.5 proof] The proof of Theorem 3.5 invokes Theorem 2.2 after checking that hat f_{overline D}(h_1,h_2) +/- hat f_D(l_1,l_2) != 2^m for distinct pairs. Theorem 2.2 requires the inequality hat f(h) +/- hat f(l) != q for the same Boolean function f. Since hat f_{overline D}(w)=2-hat f_D(w), the mixed condition checked in the proof does not imply the required condition for either f_D or f_{overline D}. The minimality assertion of Theorem 3.5 is therefore not proven, and the theorem needs either a corrected proof or a corrected statement specifying which code is claimed to be minimal.
[Section 4, Theorem 4.2 sufficient condition] Although the sufficient condition |hat f_D(beta)| < (2/3)|D| in Theorem 4.2 is valid in isolation, it is attached to an incorrect if-and-only-if statement. The paper should separate the correct sufficient condition from the false equivalence, and should re-derive the p=2 specialization of Theorem 4.1 before using it in Corollary 4.3 and in the examples that rely on the characterization.

minor comments (5)

[Section 3, Example 4] Example 4 is a p=3 example but the text describes the code as "minimal binary"; it should say "minimal ternary."
[Section 3, Theorem 3.5 statement] The notation "C_{f_D}" in the statement of Theorem 3.5 is ambiguous because the proof uses both f_D and f_{overline D}; the intended code should be stated explicitly.
[Tables IV and V] The formatting of Tables IV and V in the text is garbled, with weight columns and frequency columns not cleanly separated; this should be fixed in the final version.
[References] Reference [19] lists the year 2003 for the IMACC proceedings, but the volume number 8308 corresponds to 2013; the year should be corrected.
[Throughout] The spelling of Krawtchouk polynomials is inconsistent ("Krawchouk" vs. "Krawtchouk"); please unify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the constructions are self-contained and rely on external minimality criteria, not on their own conclusions.

full rationale

No circularity found. The Section 3 constructions are explicit: concrete defining sets are chosen, their Walsh transforms are computed directly (e.g., Lemma 3.1, Lemma 3.2, equation (9)), and minimality is checked through external criteria such as Theorem 2.2 from Ding et al. [24] or Theorem 1.2 from Heng et al. [27], not through a property of the target code that is assumed as the conclusion. The weight distributions in Tables IV–VII are derived from the computed Walsh transforms together with cited parameter formulas, and the ratio bounds w_min/w_max <= (p-1)/p follow from the resulting explicit weight lists. Theorem 4.1 is a Walsh-transform rewriting of the independent necessary-and-sufficient condition (2) of Heng et al.; it does not define minimality into its own hypothesis, and the paper does not fit any parameter to the claimed predictions. Lemma 2.3 is a self-citation of one of the authors' earlier works [32], but it is a parameter-free external lemma with stated assumptions and is used only to compute weights, not to establish minimality; under the review rules this counts as real evidence and does not raise the circularity score. The paper is self-contained against external benchmarks, and no equation or theorem reduces by construction to its own input. Even if the skeptic's sign objection to Theorem 4.2 were correct, that would be a mathematical correctness issue, not an input-output circularity issue.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The constructions rely on standard finite-field trace and Walsh-transform machinery plus a restrictive subspace condition introduced for the paper. No free parameters are fitted to data. The main fragility is the subspace condition (8) and the correctness of the quoted minimality criteria.

assumptions (4)

standard math Heng-Ding-Zhou minimality criterion (Theorem 1.2): a code C over F_p is minimal iff sum_{c in F_p^*} wt(a+cb) != (p-1)wt(a)-wt(b) for all F_p-linearly independent codewords a,b.
Used as the external criterion in Theorems 3.10, 4.1, and 4.5.
ad hoc to paper Trace-orthogonality subspace condition (8): E_i intersect E_j = {0} and E_i^perp intersect E_j^perp = {0} for all i != j.
Introduced for this construction; formula (9) and Tables IV through VII depend on it.
domain assumption The general Maiorana-McFarland template f(x,y)=<phi(x),y>+g(x) with g=1 and an injection phi from F_2^s \ U into F_2^t \ V exists for m >= 7.
Theorem 3.5 uses this class; existence of phi requires s+1 <= 2^t - 1, which holds for the stated m.
standard math Walsh-transform weight formulas for code (4): Theorem 2.1 from [24] and Lemma 2.3 from [32].
External results, though Lemma 2.3 is self-cited (Mesnager [32]).

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Pith. "Pith review of Minimal linear codes from characteristic functions." pith.science (2026). https://pith.science/paper/ZFPYMYA7

@misc{pith2026190801650,
  author       = {Pith},
  title        = {Pith review of: Minimal linear codes from characteristic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFPYMYA7}},
  note         = {Machine review of arXiv:1908.01650}
}

abstract

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb{F}_q$, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes.

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Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Minimal vectors in linear code s,

A. Ashikhmin and A. Barg, “Minimal vectors in linear code s,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 2010-2017, 19 98

work page 2010
[2]

V ariatio ns on minimal codewords in linear codes,

A. Ashikhmin, A. Barg, G. Cohen, and L. Huguet, “V ariatio ns on minimal codewords in linear codes,” in: Applied Algebr a, Algebraic Algorithms and Error-Correcting Codes (AAECC-11) (Lectur e Notes in Computer Science, vol. 948), G. Cohen, M. Giusti, a nd T. Mora, Eds. Berlin: Springer-V erlag, pp. 96-105, 1995

work page 1995
[3]

Minimal linear codes in odd cha racteristic,

D. Bartoli and M. Bonini, “Minimal linear codes in odd cha racteristic,” IEEE Trans. Inf. Theory, vol. 65, no. 7, pp. 41 52-4155, 2019

work page 2019
[4]

Minimal linear codes arising fro m blocking sets,

M. Bonini, M. Borello, “Minimal linear codes arising fro m blocking sets,” https://arxiv.gg363.site/abs/1907.04 626v1

work page 1907
[5]

Three-weight codes and association schemes,

A. R. Calderbank and J. M. Goethals, “Three-weight codes and association schemes,” Philips J. Res., vol. 39, no. 4-5, pp. 143-152, 1984

work page 1984
[6]

The geometry of two-we ight codes,

A. R. Calderbank and W. M. Kantor, “The geometry of two-we ight codes,” Bull. London Math. Soc., vol. 18, no. 2, pp. 97-1 22, 1986. 16

work page 1986
[7]

Optimized linear complementary codes implementation for hardware trojan prevention,

C. Carlet et al., “Optimized linear complementary codes implementation for hardware trojan prevention,” in Proc. 2 2nd Eur. Conf. Circuit Theory Design (ECCTD), 2015, pp. 1-4

work page 2015
[8]

Complementary dual codes for c ountermeasures to side-channel attacks,

C. Carlet and S. Guilley, “Complementary dual codes for c ountermeasures to side-channel attacks,” J. Adv. Math. Com mun., vol. 10, no. 1, pp. 131C150, 2016

work page 2016

Show all 39 references

[9]

Boolean functions for cryptography and erro r correcting codes,

C. Carlet, “Boolean functions for cryptography and erro r correcting codes,” in Boolean Models and Methods in Mathem atics, Computer Science, and Engineering, Y . Crama and P . L. Hammer, Eds. Cam bridge, U.K.: Cambridge Univ. Press, 2010, pp. 257-397

2010
[10]

Nonlinearities of S-boxes,

C. Carlet and C. Ding, “Nonlinearities of S-boxes,” Fin ite Fields and Their Applications, vol. 13, pp. 121-135, 200 7
[11]

Linear codes from highl y nonlinear functions and their secret sharing schemes,

C. Carlet, C. Ding, and J. Y uan, “Linear codes from highl y nonlinear functions and their secret sharing schemes,” IE EE Trans. Inf. Theory, vol. 51, no.6, pp. 2089-2102, 2005

2005
[12]

Four decades of research on b ent functions,

C. Carlet and S. Mesnager, “ Four decades of research on b ent functions,” Des. Codes Cryptogr., vol. 78, no. 1, pp. 5-5 0, 2016

2016
[13]

Euclidean and He rmitian LCD MDS codes,

C. Carlet, S. Mesnager, C. Tang, Y . Qi, “Euclidean and He rmitian LCD MDS codes,” Des. Codes Cryptogr., vol. 86, no. 11 , pp. 2605-2618, 2018

2018
[14]

Li near codes over Fq are equivalent to LCD codes for q > 3,

C. Carlet, S. Mesnager, C. Tang, Y . Qi, R. Pellikaan, “Li near codes over Fq are equivalent to LCD codes for q > 3,” IEEE Trans. Inf. Theory, vol. 64, no. 4, pp. 3010-3017, 2018

2018
[15]

Further properties of several classes of Boolean functions with optimum algebr aic immunity,

C. Carlet, X. Zeng, C. Li, and L. Hu, “Further properties of several classes of Boolean functions with optimum algebr aic immunity,” Des. Codes Cryptogr., vol. 52, pp. 303-338, 2009

2009
[16]

Towards secure two -party computation from the wire-tap channel,

H. Chabanne, G. Cohen, and A. Patey, “Towards secure two -party computation from the wire-tap channel,” in: Proceed ings of ICISC 2013(Lecture Notes in Computer Science, vol. 8565), H.-S. L ee and D.-G. Han Eds. Berlin: Springer-V erlag, pp. 34-46, 20 14

2013
[17]

Linear codes from simplicial co mplexes,

S. Chang and J. Y . Hyun, “Linear codes from simplicial co mplexes,” Des. Codes Cryptogr., vol. 86, no. 10, pp. 2167-21 81, 2018

2018
[18]

On bent and semi- bent quadratic Boolean functions,

P . Charpin, E. Pasalic, C. Tavernier, “On bent and semi- bent quadratic Boolean functions,” IEEE Trans. Inf. Theory , vol. 51, no. 12, pp. 4286-4298, 2005

2005
[19]

On minimal and quas i-minimal linear codes

G. Cohen, S. Mesnager, and A. Patey, “ On minimal and quas i-minimal linear codes” in: Proceedings of IMACC (Lecture N otes in Computer Science, vol. 8308), M. Stam, Eds. Berlin: Springe r-V erlag, pp. 85-98, 2003

2003
[20]

Elementary Hadamard difference sets,

J. F. Dillon, “Elementary Hadamard difference sets,” P h.D. thesis, Univ. of Maryland, 1974

1974
[21]

Linear codes from some 2-designs,

C. Ding, “Linear codes from some 2-designs,” IEEE Trans . Inf. Theory, vol. 60, no. 6, pp. 3265-3275, 2015

2015
[22]

A construction of binary linear codes from Boo lean functions,

C. Ding, “A construction of binary linear codes from Boo lean functions,” Discrete Mathematics, vol. 339, pp. 2288- 2303, 2016

2016
[23]

A Generic Construction of Cartesian Authent ication Codes,

C. Ding, T. Helleseth, T. Kl φ ve and X. Wang, “A Generic Construction of Cartesian Authent ication Codes,” IEEE Trans. Inf. Theory, vol. 53, no. 6, pp. 2229-2235, 2007

2007
[24]

Minimal binary linear cod es,

C. Ding, Z. Heng, and Z. Zhou, “Minimal binary linear cod es,” IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536 - 6545, 2018

2018
[25]

Covering and secret sharing with li near codes,

C. Ding and J. Y uan, “Covering and secret sharing with li near codes,” in: Discrete Mathematics and Theoretical Comp uter Science, Lecture Notes in Computer Science 2731, 2003, Springer V erl ag, pp. 11-25

2003
[26]

A coding theory construction of new systematic authentication codes,

C. Ding and X. Wang, “A coding theory construction of new systematic authentication codes,” Theor. Comp. Sci., vol. 330, no. 1, 81-99, 2005

2005
[27]

Minimal linear codes over ﬁnite ﬁelds,

Z. Heng, C. Ding, and Z. Zhou, “Minimal linear codes over ﬁnite ﬁelds,” Finite Fields Appl., vol. 54, pp. 176-196, 201 8
[28]

F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-C orrecting Codes, North-Holland, Amsterdam, 1977

1977
[29]

A family of difference sets in non-cyc lic groups,

R. L. McFarland, “A family of difference sets in non-cyc lic groups,” Journal of Combinatorial Theory, Series A, vol . 15, no. 1, pp. 1-10, 1973

1973
[30]

Minimal codewords and secret sharing,

J. L. Massey, “Minimal codewords and secret sharing,” i n: Proc. 6th Joint Swedish-Russian Workshop on Information Theory (Molle, Sweden, 1993), pp. 246-249

1993
[31]

Bent functions: fundamentals and result s,

S. Mesnager, “Bent functions: fundamentals and result s,” in Springer V erlag, Switzerland, 2016

2016
[32]

Linear codes with few weights from weakly regular bent functions based on a generic construction,

S. Mesnager, “Linear codes with few weights from weakly regular bent functions based on a generic construction,” Cr yptogr. Commun., vol. 9, pp. 71-84, 2017

2017
[33]

Complementary dual algebr aic geometry codes,

S. Mesnager, C. Tang, Y . Qi, “Complementary dual algebr aic geometry codes,” IEEE Trans. Inf. Theory, vol. 64, no. 4, pp. 2390-2397, 2018

2018
[34]

Linear codes from we akly regular plateaued functions and their secret sharing s chemes,

S. Mesnager, F. Ozbudak, A. Sınak, “Linear codes from we akly regular plateaued functions and their secret sharing s chemes,” Des. Codes Cryptogr., vol. 87, no. 2-3, pp. 463-480, 2019

2019
[35]

Upp er and lower bounds on maximum nonlinearity of n-input m-out put Boolean function,

T. Wadayama, T. Hada, K. Wakasugi, and M. Kasahara, “Upp er and lower bounds on maximum nonlinearity of n-input m-out put Boolean function,” Des. Codes Cryptogr., vol. 23, pp. 23-33 , 2001

2001
[36]

Three classes of minimal linear codes over the ﬁnite ﬁelds of odd characteristic,

G. Xu, L. Qu, “Three classes of minimal linear codes over the ﬁnite ﬁelds of odd characteristic,” IEEE Trans. Inf. The ory, vol. 65, no. 11, pp. 7067-7078, 2019

2019
[37]

Secret sharing schemes from three c lasses of linear codes,

J. Y uan and C. Ding, “Secret sharing schemes from three c lasses of linear codes,” IEEE Trans. Inf. Theory, vol. 52, no . 1, pp. 206-212, 2006

2006
[38]

Four families of minimal bi nary linear codes with wmin/w max ≤ 1/ 2,

W. Zhang, H. Yan, and H. Wei, “Four families of minimal bi nary linear codes with wmin/w max ≤ 1/ 2,” Appl. Algerbra Eng. Com- mun. Comput., vol. 30, no. 2, pp. 175-184, 2019

2019
[39]

Binary LCD codes and sel f-orthogonal codes from a generic construction,

Z. Zhou, X. Li, C. Tang, C. Ding, “Binary LCD codes and sel f-orthogonal codes from a generic construction,” IEEE Tran s. Inf. Theory, vol. 65, no. 1, pp. 16-27, 2019

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Minimal vectors in linear code s,

A. Ashikhmin and A. Barg, “Minimal vectors in linear code s,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 2010-2017, 19 98

work page 2010

[2] [2]

V ariatio ns on minimal codewords in linear codes,

A. Ashikhmin, A. Barg, G. Cohen, and L. Huguet, “V ariatio ns on minimal codewords in linear codes,” in: Applied Algebr a, Algebraic Algorithms and Error-Correcting Codes (AAECC-11) (Lectur e Notes in Computer Science, vol. 948), G. Cohen, M. Giusti, a nd T. Mora, Eds. Berlin: Springer-V erlag, pp. 96-105, 1995

work page 1995

[3] [3]

Minimal linear codes in odd cha racteristic,

D. Bartoli and M. Bonini, “Minimal linear codes in odd cha racteristic,” IEEE Trans. Inf. Theory, vol. 65, no. 7, pp. 41 52-4155, 2019

work page 2019

[4] [4]

Minimal linear codes arising fro m blocking sets,

M. Bonini, M. Borello, “Minimal linear codes arising fro m blocking sets,” https://arxiv.gg363.site/abs/1907.04 626v1

work page 1907

[5] [5]

Three-weight codes and association schemes,

A. R. Calderbank and J. M. Goethals, “Three-weight codes and association schemes,” Philips J. Res., vol. 39, no. 4-5, pp. 143-152, 1984

work page 1984

[6] [6]

The geometry of two-we ight codes,

A. R. Calderbank and W. M. Kantor, “The geometry of two-we ight codes,” Bull. London Math. Soc., vol. 18, no. 2, pp. 97-1 22, 1986. 16

work page 1986

[7] [7]

Optimized linear complementary codes implementation for hardware trojan prevention,

C. Carlet et al., “Optimized linear complementary codes implementation for hardware trojan prevention,” in Proc. 2 2nd Eur. Conf. Circuit Theory Design (ECCTD), 2015, pp. 1-4

work page 2015

[8] [8]

Complementary dual codes for c ountermeasures to side-channel attacks,

C. Carlet and S. Guilley, “Complementary dual codes for c ountermeasures to side-channel attacks,” J. Adv. Math. Com mun., vol. 10, no. 1, pp. 131C150, 2016

work page 2016