Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding

Ilaria Cardinali; Luca Giuzzi

arxiv: 2507.16694 · v1 · submitted 2025-07-22 · 🧮 math.CO · cs.IT· math.IT

Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding

Ilaria Cardinali , Luca Giuzzi This is my paper

Pith reviewed 2026-05-19 03:23 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT

keywords linear codesprojective geometrytwisted embeddingsfinite fieldsautomorphism groupsminimum distancepoint-hyperplane geometryminimal codes

0 comments

The pith

The twisted embedding of the point-hyperplane geometry produces a minimal linear code whose parameters, minimum distance, and automorphism group follow from the geometry.

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

This paper examines the linear codes obtained by applying a twisted embedding to the point-hyperplane geometry of a projective space over a finite field. An automorphism σ of the field is used to define the map that sends each incident point-hyperplane pair to a pure tensor x^σ ⊗ ξ in the tensor product space. The resulting projective system Λ_σ generates a linear code C(Λ_σ) that the authors prove is minimal. They compute the code parameters and minimum distance, identify the automorphism group, give a geometrical description of the lowest and second-lowest weight codewords, and determine the maximum weight when both the field order q and dimension parameter n are odd.

Core claim

For σ not equal to 1, the code C(Λ_σ) arising from the projective system Λ_σ = ε_σ(Γ-bar), where the twisted embedding ε_σ sends an incident pair ([x], [ξ]) to the projective point of the pure tensor x^σ ⊗ ξ with ξ(x) = 0, is a minimal code. Its parameters and minimum distance are determined, its automorphism group is identified, the minimum and second-lowest weight codewords are characterized geometrically, and the maximum weight is found when q and n are both odd.

What carries the argument

The twisted embedding ε_σ of the point-hyperplane geometry, which maps incident pairs to points in PG(V ⊗ V*) and generates the projective system Λ_σ whose linear span is the code C(Λ_σ).

If this is right

The code C(Λ_σ) is minimal.
Its parameters and minimum distance are explicitly determined from the geometry.
Its automorphism group is identified.
The minimum and second-lowest weight codewords receive a geometrical characterization.
The maximum weight is determined when q and n are both odd.

Where Pith is reading between the lines

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

The twisted construction provides a direct comparison to the Segre embedding case treated in the companion paper for σ = 1, highlighting how the field automorphism alters the resulting code properties.
The geometrical characterizations of low-weight words may extend to decoding procedures or to similar incidence geometries beyond projective spaces.
These codes form a family whose weight distributions could be contrasted with other geometrically defined codes to identify cases with favorable distance properties.

Load-bearing premise

The twisted map ε_σ yields a well-defined projective system from the point-hyperplane incidence structure so that the code's weight distribution and automorphism group can be read off from the original geometry.

What would settle it

An explicit weight enumeration for small q and n that shows a codeword weight outside the stated minimum distance or maximum weight (when q and n are odd) or a codeword whose support fails the geometrical characterization would disprove the claims.

read the original abstract

Let $\bar{\Gamma}$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $\sigma$ is an automorphism of $\mathbb{F}_q$ and consider the projective embedding $\varepsilon_{\sigma}$ of $\bar{\Gamma}$ into the projective space $\mathrm{PG}(V\otimes V^*)$ mapping the point $([x],[\xi])\in \bar{\Gamma}$ to the projective point represented by the pure tensor $x^{\sigma}\otimes \xi$, with $\xi(x)=0.$ In [I. Cardinali, L. Giuzzi, Linear codes arising from the point-hyperplane geometry -- part I: the Segre embedding (Jun. 2025). arXiv:2506.21309, doi:10.48550/ARXIV.2506.21309] we focused on the case $\sigma=1$ and we studied the projective code arising from the projective system $\Lambda_1=\varepsilon_{1}(\bar{\Gamma}).$ Here we focus on the case $\sigma\not=1$ and we investigate the linear code ${\mathcal C}(\Lambda_{\sigma})$ arising from the projective system $\Lambda_{\sigma}=\varepsilon_{\sigma}(\bar{\Gamma}).$ In particular, after having verified that $\mathcal{C}( \Lambda_{\sigma})$ is a minimal code, we determine its parameters, its minimum distance as well as its automorphism group. We also give a (geometrical) characterization of its minimum and second lowest weight codewords and determine its maximum weight when $q$ and $n$ are both odd.

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

This extends the Segre case to twisted embeddings, giving explicit parameters, minimum distance, automorphism group, and weight characterizations for the resulting minimal codes.

read the letter

This paper is the follow-up to the authors' Part I, moving from the standard Segre embedding to the twisted version using a field automorphism σ not equal to the identity. They show that the code coming from the image of the point-hyperplane geometry under this twisted map is minimal, and they work out its length, dimension, minimum distance, and automorphism group. They also describe the codewords of minimum and second minimum weight in geometric terms, and they pin down the maximum weight in the case where q and n are both odd. The new part is the treatment of the twisted case. It takes fresh work to figure out the automorphism group and to characterize the weights, because the twisting changes how the points sit in the tensor product space. They stick to the same geometric setup as before, which makes the extension coherent. What they do well is deliver explicit data for an infinite family of these codes. In this corner of coding theory, having the parameters and the group action spelled out is helpful for anyone who might want to use or compare these constructions. The main thing to watch is whether the twisted map really gives distinct points. The concern is that different incident pairs could land on the same projective point after twisting, which would mean repeated coordinates in the code and change the weight distribution. Since the abstract says they verified minimality, they must have shown the points are all distinct or handled any overlaps. A referee should check that step carefully because everything else rests on it. The fact that the maximum weight is only settled for odd q and n is a minor loose end. This paper is for specialists in finite geometry and algebraic coding theory. Anyone studying minimal linear codes or codes with large automorphism groups from geometric sources would get something out of the explicit results here. It deserves a serious referee. The claims are specific, the extension requires real work, and the geometric arguments are the kind that can be checked. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper examines linear codes C(Λ_σ) arising from the twisted embedding ε_σ of the point-hyperplane geometry of PG(V) (dim V = n+1) into PG(V ⊗ V*), where σ is a non-identity automorphism of F_q. The authors verify that C(Λ_σ) is minimal, determine its parameters, minimum distance and automorphism group, give a geometric characterization of minimum and second-lowest weight codewords, and compute the maximum weight when q and n are both odd.

Significance. If the claims hold, the work supplies an explicit family of minimal codes with known parameters, distance, and automorphism group obtained from a natural twisted geometric construction. The geometric characterizations of low-weight words are a concrete strength, extending the Segre case treated in Part I and providing falsifiable predictions about weight distributions that can be checked against the incidence structure.

major comments (2)

[Definition of the twisted embedding ε_σ and construction of Λ_σ] The verification that C(Λ_σ) is minimal and the subsequent parameter, distance, and weight-distribution claims all presuppose that Λ_σ is a simple projective system (distinct points, no multiplicities). The scaling argument in the definition of ε_σ (x^σ ⊗ ξ scaled by λ^σ or μ) does not automatically exclude collisions x^σ ⊗ ξ = λ y^σ ⊗ η for distinct incident pairs ([x],[ξ]) ≠ ([y],[η]). A self-contained proof that ε_σ is injective on the set of incident pairs is required; without it the minimality statement and all derived quantities rest on an unverified assumption.
[Automorphism group computation] The computation of the automorphism group of C(Λ_σ) is stated to follow from the geometry of the original point-hyperplane incidence structure. The precise statement of which automorphisms of PG(V) lift to automorphisms of the code (and whether the full group is induced or larger) should be isolated in a dedicated proposition or theorem, with an explicit comparison to the σ = id case of Part I.

minor comments (2)

[Notation and preliminaries] Notation for projective points and hyperplanes should be uniform (e.g., consistent use of [x] versus x throughout).
[References] The reference to Part I should include the arXiv identifier in the bibliography for easy cross-checking.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The comments are helpful for improving the clarity and rigor of the manuscript. We address each major comment below.

read point-by-point responses

Referee: [Definition of the twisted embedding ε_σ and construction of Λ_σ] The verification that C(Λ_σ) is minimal and the subsequent parameter, distance, and weight-distribution claims all presuppose that Λ_σ is a simple projective system (distinct points, no multiplicities). The scaling argument in the definition of ε_σ (x^σ ⊗ ξ scaled by λ^σ or μ) does not automatically exclude collisions x^σ ⊗ ξ = λ y^σ ⊗ η for distinct incident pairs ([x],[ξ]) ≠ ([y],[η]). A self-contained proof that ε_σ is injective on the set of incident pairs is required; without it the minimality statement and all derived quantities rest on an unverified assumption.

Authors: We agree that a self-contained proof of injectivity is required to confirm that Λ_σ forms a simple projective system without multiplicities. In the revised version we will insert a dedicated lemma immediately after the definition of ε_σ. The lemma will explicitly rule out collisions by considering the two possible scalings (λ^σ and μ) and showing that if x^σ ⊗ ξ = λ y^σ ⊗ η for incident pairs ([x],[ξ]) ≠ ([y],[η]), then the pairs must coincide, using the incidence condition ξ(x)=0=η(y) and the fact that σ is a field automorphism. This will underpin the minimality claim and all derived parameters. revision: yes
Referee: [Automorphism group computation] The computation of the automorphism group of C(Λ_σ) is stated to follow from the geometry of the original point-hyperplane incidence structure. The precise statement of which automorphisms of PG(V) lift to automorphisms of the code (and whether the full group is induced or larger) should be isolated in a dedicated proposition or theorem, with an explicit comparison to the σ = id case of Part I.

Authors: We accept the suggestion to improve the presentation. In the revision we will extract the automorphism-group result into a standalone theorem (placed after the minimum-distance section). The theorem will state precisely which collineations of PG(V) extend to automorphisms of C(Λ_σ), indicate whether the full group is induced by the geometry, and include a direct side-by-side comparison with the corresponding result for the Segre embedding (σ = id) treated in Part I, highlighting the additional constraints imposed by the twisting automorphism σ. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the twisted embedding ε_σ explicitly via the pure tensor map on incident point-hyperplane pairs and constructs the projective system Λ_σ and code C(Λ_σ) directly from it. Parameters, minimum distance, automorphism group, and weight characterizations are obtained by analyzing the incidence geometry of the original point-hyperplane structure under this map. The citation to the authors' Part I applies only to the untwisted Segre case (σ = id) and supplies background rather than load-bearing justification for the twisted results. No step equates a derived quantity to a fitted parameter, renames a prior result, or reduces the central claims to a self-citation chain by construction. The work is a standard mathematical derivation relying on linear algebra and finite geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper relies on standard facts about projective spaces over finite fields and the definition of the twisted embedding. No free parameters, new axioms, or invented entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5862 in / 1150 out tokens · 31131 ms · 2026-05-19T03:23:50.105628+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the twisted embedding ε_σ of Γ̄ into PG(V ⊗ V*) mapping ([x],[ξ]) to the projective point represented by the pure tensor x^σ ⊗ ξ, with ξ(x)=0
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

after having verified that C(Λ_σ) is a minimal code, we determine its parameters, its minimum distance as well as its automorphism group

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

[1]

G. N. Alfarano, M. Borello, A. Neri, A geometric characterization of minimal codes and their asymptotic performance, Advances in Mathematics of Commu- nications 16 (1) (2022) 115–133. doi:10.3934/amc.2020104. 26

work page doi:10.3934/amc.2020104 2022
[2]

G. N. Alfarano, M. Borello, A. Neri, A. Ravagnani, Three combinatorial perspec- tives on minimal codes, SIAM Journal on Discrete Mathematics 36 (1) (2022) 461–489. doi:10.1137/21M1391493

work page doi:10.1137/21m1391493 2022
[3]

Ashikhmin, A

A. Ashikhmin, A. Barg, Minimal vectors in linear codes, Institute of Electrical and Electronics Engineers. Transactions on Information Theory 44 (5) (1998) 2010–2017. doi:10.1109/18.705584

work page doi:10.1109/18.705584 1998
[4]

Beelen, S

P. Beelen, S. R. Ghorpade, S. U. Hasan, Linear codes associated to deter- minantal varieties, Discrete Mathematics 338 (8) (2015) 1493–1500. doi: 10.1016/j.disc.2015.03.009

work page doi:10.1016/j.disc.2015.03.009 2015
[5]

Bonini, M

M. Bonini, M. Borello, Minimal linear codes arising from blocking sets, Journal of Algebraic Combinatorics. An International Journal 53 (2) (2021) 327–341. doi:10.1007/s10801-019-00930-6

work page doi:10.1007/s10801-019-00930-6 2021
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Cardinali, L

I. Cardinali, L. Giuzzi, On minimal codes arising from projective embeddings of point-line geometries., in preparation

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The relatively universal cover of the natural embedding of the long root geometry for the group $\mathrm{SL}(n+1,\mathbb{K})$

I. Cardinali, L. Giuzzi, A. Pasini, The relatively universal cover of the natural embedding of the long root geometry for the group SL( n + 1, K), preprint (Oct. 2024). arXiv:2410.17023, doi:10.48550/ARXIV.2410.17023

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2410.17023 2024
[8]

Cardinali, L

I. Cardinali, L. Giuzzi, Minimum distance of symplectic Grassmann codes, Lin- ear Algebra and its Applications 488 (2016) 124–134. doi:10.1016/j.laa. 2015.09.031

work page doi:10.1016/j.laa 2016
[9]

Cardinali, L

I. Cardinali, L. Giuzzi, Line Hermitian Grassmann codes and their parameters, Finite Fields and their Applications 51 (2018) 407–432. doi:10.1016/j.ffa. 2018.02.006

work page doi:10.1016/j.ffa 2018
[10]

Cardinali, L

I. Cardinali, L. Giuzzi, Minimum distance of orthogonal line-Grassmann codes in even characteristic, Journal of Pure and Applied Algebra 222 (10) (2018) 2975–2988. doi:10.1016/j.jpaa.2017.11.009

work page doi:10.1016/j.jpaa.2017.11.009 2018
[11]

Cardinali, L

I. Cardinali, L. Giuzzi, Linear codes arising from the point-hyperplane geometry — part I: the Segre embedding (Jun. 2025).arXiv:2506.21309, doi:10.48550/ ARXIV.2506.21309

work page arXiv 2025
[12]

Cardinali, L

I. Cardinali, L. Giuzzi, K. V. Kaipa, A. Pasini, Line polar Grassmann codes of orthogonal type, Journal of Pure and Applied Algebra 220 (5) (2016) 1924–1934. doi:10.1016/j.jpaa.2015.10.007. 27

work page doi:10.1016/j.jpaa.2015.10.007 2016
[13]

A. A. Davydov, M. Giulietti, S. Marcugini, F. Pambianco, Linear nonbinary cov- ering codes and saturating sets in projective spaces, Advances in Mathematics of Communications 5 (1) (2011) 119–147. doi:10.3934/amc.2011.5.119

work page doi:10.3934/amc.2011.5.119 2011
[14]

W. M. Kantor, Subgroups of classical groups generated by long root elements, Transactions of the American Mathematical Society 248 (2) (1979) 347–379. doi:10.2307/1998972

work page doi:10.2307/1998972 1979
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F. J. MacWilliams, N. J. A. Sloane, The theory of error-correcting codes. Vol. 16 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977

work page 1977
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F. J. MacWilliams, Combinatorial properties of elementary abelian groups, Har- vard University, Cambridge, MA, 1962, thesis (Ph.D.)–Radcliffe College

work page 1962
[17]

Massey, Minimal codewords and secret sharing, in: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, 1993, pp

J. Massey, Minimal codewords and secret sharing, in: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, 1993, pp. 276–279

work page 1993
[18]

Pasini, Embeddings and hyperplanes of the Lie geometry An,{1,n}(F), Comb

A. Pasini, Embeddings and hyperplanes of the Lie geometry An,{1,n}(F), Comb. Theory 4 (2) (2024) Paper No. 5, 25

work page 2024
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Pasini, Geometric hyperplanes of the Lie geometry An,{1,n}(F), Ricerche di Matematica (Apr

A. Pasini, Geometric hyperplanes of the Lie geometry An,{1,n}(F), Ricerche di Matematica (Apr. 2024). doi:10.1007/s11587-024-00859-4

work page doi:10.1007/s11587-024-00859-4 2024
[20]

C. Tang, Y. Qiu, Q. Liao, Z. Zhou, Full characterization of minimal linear codes as cutting blocking sets, Institute of Electrical and Electronics Engineers. Transactions on Information Theory 67 (6) (2021) 3690–3700. doi:10.1109/ TIT.2021.3070377

work page arXiv 2021
[21]

Tsfasman, S

M. Tsfasman, S. Vl˘ adut ¸, D. Nogin, Algebraic geometric codes: basic notions, Vol. 139, American Mathematical Society, 2007. doi:10.1090/surv/139

work page doi:10.1090/surv/139 2007
[22]

Van Maldeghem, Hyperplanes of Segre geometries, Ars Combin

H. Van Maldeghem, Hyperplanes of Segre geometries, Ars Combin. 160 (2024) 59–71. doi:10.61091/ars-160-07. Authors’ addresses: Ilaria Cardinali, Dep. Information Engineering and Mathematics University of Siena Via Roma 56, I-53100 Siena, Italy ilaria.cardinali@unisi.it Luca Giuzzi D.I.C.A.T.A.M. Universit` a di Brescia Via Branze 43, I-25123 Brescia, Italy...

work page doi:10.61091/ars-160-07 2024

[1] [1]

G. N. Alfarano, M. Borello, A. Neri, A geometric characterization of minimal codes and their asymptotic performance, Advances in Mathematics of Commu- nications 16 (1) (2022) 115–133. doi:10.3934/amc.2020104. 26

work page doi:10.3934/amc.2020104 2022

[2] [2]

G. N. Alfarano, M. Borello, A. Neri, A. Ravagnani, Three combinatorial perspec- tives on minimal codes, SIAM Journal on Discrete Mathematics 36 (1) (2022) 461–489. doi:10.1137/21M1391493

work page doi:10.1137/21m1391493 2022

[3] [3]

Ashikhmin, A

A. Ashikhmin, A. Barg, Minimal vectors in linear codes, Institute of Electrical and Electronics Engineers. Transactions on Information Theory 44 (5) (1998) 2010–2017. doi:10.1109/18.705584

work page doi:10.1109/18.705584 1998

[4] [4]

Beelen, S

P. Beelen, S. R. Ghorpade, S. U. Hasan, Linear codes associated to deter- minantal varieties, Discrete Mathematics 338 (8) (2015) 1493–1500. doi: 10.1016/j.disc.2015.03.009

work page doi:10.1016/j.disc.2015.03.009 2015

[5] [5]

Bonini, M

M. Bonini, M. Borello, Minimal linear codes arising from blocking sets, Journal of Algebraic Combinatorics. An International Journal 53 (2) (2021) 327–341. doi:10.1007/s10801-019-00930-6

work page doi:10.1007/s10801-019-00930-6 2021

[6] [6]

Cardinali, L

I. Cardinali, L. Giuzzi, On minimal codes arising from projective embeddings of point-line geometries., in preparation

work page

[7] [7]

The relatively universal cover of the natural embedding of the long root geometry for the group $\mathrm{SL}(n+1,\mathbb{K})$

I. Cardinali, L. Giuzzi, A. Pasini, The relatively universal cover of the natural embedding of the long root geometry for the group SL( n + 1, K), preprint (Oct. 2024). arXiv:2410.17023, doi:10.48550/ARXIV.2410.17023

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2410.17023 2024

[8] [8]

Cardinali, L

I. Cardinali, L. Giuzzi, Minimum distance of symplectic Grassmann codes, Lin- ear Algebra and its Applications 488 (2016) 124–134. doi:10.1016/j.laa. 2015.09.031

work page doi:10.1016/j.laa 2016

[9] [9]

Cardinali, L

I. Cardinali, L. Giuzzi, Line Hermitian Grassmann codes and their parameters, Finite Fields and their Applications 51 (2018) 407–432. doi:10.1016/j.ffa. 2018.02.006

work page doi:10.1016/j.ffa 2018

[10] [10]

Cardinali, L

I. Cardinali, L. Giuzzi, Minimum distance of orthogonal line-Grassmann codes in even characteristic, Journal of Pure and Applied Algebra 222 (10) (2018) 2975–2988. doi:10.1016/j.jpaa.2017.11.009

work page doi:10.1016/j.jpaa.2017.11.009 2018

[11] [11]

Cardinali, L

I. Cardinali, L. Giuzzi, Linear codes arising from the point-hyperplane geometry — part I: the Segre embedding (Jun. 2025).arXiv:2506.21309, doi:10.48550/ ARXIV.2506.21309

work page arXiv 2025

[12] [12]

Cardinali, L

I. Cardinali, L. Giuzzi, K. V. Kaipa, A. Pasini, Line polar Grassmann codes of orthogonal type, Journal of Pure and Applied Algebra 220 (5) (2016) 1924–1934. doi:10.1016/j.jpaa.2015.10.007. 27

work page doi:10.1016/j.jpaa.2015.10.007 2016

[13] [13]

A. A. Davydov, M. Giulietti, S. Marcugini, F. Pambianco, Linear nonbinary cov- ering codes and saturating sets in projective spaces, Advances in Mathematics of Communications 5 (1) (2011) 119–147. doi:10.3934/amc.2011.5.119

work page doi:10.3934/amc.2011.5.119 2011

[14] [14]

W. M. Kantor, Subgroups of classical groups generated by long root elements, Transactions of the American Mathematical Society 248 (2) (1979) 347–379. doi:10.2307/1998972

work page doi:10.2307/1998972 1979

[15] [15]

F. J. MacWilliams, N. J. A. Sloane, The theory of error-correcting codes. Vol. 16 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977

work page 1977

[16] [16]

F. J. MacWilliams, Combinatorial properties of elementary abelian groups, Har- vard University, Cambridge, MA, 1962, thesis (Ph.D.)–Radcliffe College

work page 1962

[17] [17]

Massey, Minimal codewords and secret sharing, in: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, 1993, pp

J. Massey, Minimal codewords and secret sharing, in: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, 1993, pp. 276–279

work page 1993

[18] [18]

Pasini, Embeddings and hyperplanes of the Lie geometry An,{1,n}(F), Comb

A. Pasini, Embeddings and hyperplanes of the Lie geometry An,{1,n}(F), Comb. Theory 4 (2) (2024) Paper No. 5, 25

work page 2024

[19] [19]

Pasini, Geometric hyperplanes of the Lie geometry An,{1,n}(F), Ricerche di Matematica (Apr

A. Pasini, Geometric hyperplanes of the Lie geometry An,{1,n}(F), Ricerche di Matematica (Apr. 2024). doi:10.1007/s11587-024-00859-4

work page doi:10.1007/s11587-024-00859-4 2024

[20] [20]

C. Tang, Y. Qiu, Q. Liao, Z. Zhou, Full characterization of minimal linear codes as cutting blocking sets, Institute of Electrical and Electronics Engineers. Transactions on Information Theory 67 (6) (2021) 3690–3700. doi:10.1109/ TIT.2021.3070377

work page arXiv 2021

[21] [21]

Tsfasman, S

M. Tsfasman, S. Vl˘ adut ¸, D. Nogin, Algebraic geometric codes: basic notions, Vol. 139, American Mathematical Society, 2007. doi:10.1090/surv/139

work page doi:10.1090/surv/139 2007

[22] [22]

Van Maldeghem, Hyperplanes of Segre geometries, Ars Combin

H. Van Maldeghem, Hyperplanes of Segre geometries, Ars Combin. 160 (2024) 59–71. doi:10.61091/ars-160-07. Authors’ addresses: Ilaria Cardinali, Dep. Information Engineering and Mathematics University of Siena Via Roma 56, I-53100 Siena, Italy ilaria.cardinali@unisi.it Luca Giuzzi D.I.C.A.T.A.M. Universit` a di Brescia Via Branze 43, I-25123 Brescia, Italy...

work page doi:10.61091/ars-160-07 2024