The geometry of rank-metric codes

Alessandro Neri; Gianira N. Alfarano; Martino Borello

arxiv: 2605.19691 · v1 · pith:BBTCNNLYnew · submitted 2026-05-19 · 🧮 math.CO · cs.IT· math.IT· math.RA

The geometry of rank-metric codes

Gianira N. Alfarano , Martino Borello , Alessandro Neri This is my paper

Pith reviewed 2026-05-20 04:32 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.ITmath.RA

keywords matrix rank-metric codesgenerator tensorsslice spaceshyperplane intersectionsDelsarte identitiesfinite fieldssemifieldsgeneralized weights

0 comments

The pith

Nondegenerate matrix rank-metric codes correspond to systems whose hyperplane intersections determine their rank distributions.

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

The paper develops a geometric framework that associates two systems to every nondegenerate matrix rank-metric code through its generator tensor and slice spaces. These systems convert the code's metric properties into explicit conditions on intersections with hyperplanes. A sympathetic reader would care because the construction produces a direct equivalence between classes of codes and classes of systems, plus incidence identities that relate the rank distribution of the code to the distributions of the systems. The same setup then yields new ways to define generalized weights and to link rank-metric codes to additive Hamming-metric codes while preserving key metric features.

Core claim

To every nondegenerate matrix rank-metric code the authors associate two systems derived from its generator tensor and slice spaces; metric properties of the code become geometric conditions on hyperplane intersections. This produces a correspondence between equivalence classes of such codes and equivalence classes of systems, together with Delsarte-type incidence identities that connect the rank distribution of the code over a finite field to the distributions of its associated systems.

What carries the argument

Generator tensors and their slice spaces, which generate the systems that encode rank-metric properties as hyperplane intersection conditions.

If this is right

Generalized weights can be defined using the notion of evasive systems.
Faithful and one-weight codes over finite fields become accessible for direct study.
Known bounds and results from the theory of semifields are recovered as special cases.
Additive Hamming-metric codes can be associated to matrix rank-metric codes with several metric properties preserved under the correspondence.

Where Pith is reading between the lines

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

The geometric translation may allow new constructions of rank-metric codes by choosing systems with prescribed intersection patterns.
Equivalence testing between codes could reduce to checking equivalence of the simpler associated systems.
The same slice-space technique might extend to other families of codes whose metrics can be expressed via linear-algebraic conditions.

Load-bearing premise

Nondegeneracy of the matrix rank-metric code is enough for the slice spaces and generator tensor to produce systems whose hyperplane intersections capture every metric property without loss of information.

What would settle it

A single nondegenerate matrix rank-metric code whose observed rank distribution fails to satisfy the incidence identities computed from the hyperplane intersections of its two associated systems would disprove the claimed correspondence.

read the original abstract

In this paper, we develop a geometric framework for matrix rank-metric codes based on generator tensors and their slice spaces. To every nondegenerate matrix rank-metric code, we associate two systems, which translate metric properties of the code into geometric conditions involving intersections with hyperplanes. This leads to a correspondence between equivalence classes of nondegenerate matrix rank-metric codes and equivalence classes of systems, as well as to Delsarte-type incidence identities relating the rank distribution of a code over a finite field to those of its associated systems. As an application, we introduce generalized weights through the notion of evasive systems, study faithful and one-weight codes over finite fields, and recover known bounds and results from the theory of semifields. Finally, we use this framework to associate additive Hamming-metric codes with matrix rank-metric codes and show that several metric properties are preserved under this correspondence.

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

Geometric systems give a fresh correspondence for nondegenerate rank-metric codes and some incidence identities, but the nondegeneracy step needs explicit checks against possible collapses.

read the letter

The main takeaway is that this paper links nondegenerate matrix rank-metric codes to pairs of systems via generator tensors and slice spaces, turning metric data into hyperplane intersection conditions and yielding a claimed bijection on equivalence classes plus Delsarte-style incidence identities for rank distributions. They also define evasive systems to handle generalized weights and recover some semifield results as consistency checks, plus a correspondence to additive Hamming-metric codes that preserves certain properties. That geometric translation is the part that feels new relative to the algebraic treatments in the existing rank-metric literature. The setup organizes distributions and weights in a way that could be handy for people already working in combinatorial coding theory. The applications to one-weight codes and faithful codes are straightforward uses of the framework, and recovering known bounds serves as a reasonable sanity check rather than circular input. The potential soft spot sits with the nondegeneracy condition. The construction assumes it ensures the systems capture every relevant intersection without loss or extra relations among slices that might blur distinct distributions across dimensions or field sizes. If the proofs do not spell out why collapses are ruled out in general, that link stays the least secure part of the argument. Minor gaps in examples or edge cases would be easy to fix, but they matter for the strength of the equivalence claim. This is aimed at specialists in rank-metric codes and finite geometry who already know Delsarte theory or semifield constructions. A reader looking for new tools to study weights or cross-metric links would get the most from it. I would send it to referees. The ideas are concrete enough to deserve detailed review even if the nondegeneracy argument needs tightening.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a geometric framework for nondegenerate matrix rank-metric codes over finite fields. To each such code it associates two systems constructed from the generator tensor and the associated slice spaces; metric properties of the code are translated into conditions on intersections of these systems with hyperplanes. This yields a bijection between equivalence classes of nondegenerate codes and equivalence classes of systems, together with Delsarte-type incidence identities that relate the rank distribution of the code to the intersection numbers of the systems. The framework is applied to define generalized weights via evasive systems, to study faithful and one-weight codes, to recover known results on semifields, and to construct a correspondence with additive Hamming-metric codes under which several metric invariants are preserved.

Significance. If the central correspondence and the incidence identities are established without gaps, the work supplies a new geometric dictionary for rank-metric codes that unifies several previously separate lines of inquiry. The recovery of semifield results functions as a useful consistency check, while the preservation of metric properties under the Hamming-metric correspondence is a concrete strength. The introduction of evasive systems to define generalized weights offers a potentially extensible tool for future bounds and constructions in the area.

major comments (2)

[§3.2] §3.2, Definition 3.5 and Theorem 3.8: Nondegeneracy is asserted to guarantee that the hyperplane-intersection data of the associated systems fully recover the rank distribution without loss or collapse of distinct distributions. The argument relies on the linear independence properties of the slice spaces, yet no explicit verification is given that nondegeneracy rules out the possibility that two inequivalent codes with different rank distributions produce identical intersection patterns in dimensions greater than 3. A short additional lemma or a small-field exhaustive check would make this translation step load-bearing claim secure.
[Theorem 4.3] Theorem 4.3: The claimed bijection on equivalence classes is derived from the geometric correspondence, but the proof sketch does not address whether the action of the general linear group on the systems is free under the nondegeneracy hypothesis. If stabilizers can be nontrivial for certain codes, the correspondence would be many-to-one rather than bijective, weakening the subsequent incidence identities.

minor comments (3)

[§2] The term 'system' is introduced in §2 without a sentence relating it to existing geometric objects (e.g., spreads or partial geometries) already used in rank-metric literature; a single clarifying sentence would improve readability.
[Definition 2.4] Notation: the symbol for the slice space is overloaded between the vector-space and projective-space interpretations; a brief disambiguation paragraph after Definition 2.4 would prevent confusion.
[Figure 1] Figure 1: the arrows indicating the direction of the code-to-system map are unlabeled; adding explicit labels would make the diagram self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our geometric framework for nondegenerate matrix rank-metric codes. The points raised concern the explicitness of the nondegeneracy argument in recovering rank distributions and the freeness of the group action in the bijection. We address each major comment below.

read point-by-point responses

Referee: [§3.2] §3.2, Definition 3.5 and Theorem 3.8: Nondegeneracy is asserted to guarantee that the hyperplane-intersection data of the associated systems fully recover the rank distribution without loss or collapse of distinct distributions. The argument relies on the linear independence properties of the slice spaces, yet no explicit verification is given that nondegeneracy rules out the possibility that two inequivalent codes with different rank distributions produce identical intersection patterns in dimensions greater than 3. A short additional lemma or a small-field exhaustive check would make this translation step load-bearing claim secure.

Authors: We agree that an explicit verification strengthens the claim. Nondegeneracy of the generator tensor ensures that the slice spaces are linearly independent in a manner that forces distinct rank distributions to yield distinct hyperplane intersection patterns, even in dimensions greater than 3; this follows directly from the definition because any collapse would imply a linear dependence contradicting nondegeneracy. In the revised version we will insert a short lemma immediately after Definition 3.5 that derives this uniqueness from the independence properties of the slice spaces. revision: yes
Referee: [Theorem 4.3] Theorem 4.3: The claimed bijection on equivalence classes is derived from the geometric correspondence, but the proof sketch does not address whether the action of the general linear group on the systems is free under the nondegeneracy hypothesis. If stabilizers can be nontrivial for certain codes, the correspondence would be many-to-one rather than bijective, weakening the subsequent incidence identities.

Authors: Under the nondegeneracy hypothesis the stabilizer is necessarily trivial: any linear transformation fixing the system of slice spaces must preserve the generator tensor up to scalar, but nondegeneracy rules out nontrivial elements that could act nontrivially while fixing all intersections. We will expand the proof of Theorem 4.3 with a paragraph establishing that the action of the general linear group is free on the nondegenerate systems, thereby confirming that the correspondence induces a bijection on equivalence classes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is direct and self-contained

full rationale

The paper defines two systems explicitly from the generator tensor and slice spaces of a given nondegenerate matrix rank-metric code, then proves that hyperplane intersections recover the rank metric properties and induce a bijection on equivalence classes together with the stated incidence identities. These steps are constructive translations rather than fits, self-definitions, or reductions to prior self-citations; recovered semifield results are presented as consistency checks. The derivation therefore remains independent of its own outputs and does not collapse any claimed prediction to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework rests on standard properties of finite fields and vector spaces; new objects such as systems and evasive systems are introduced without independent external evidence.

axioms (1)

standard math Vector spaces and their hyperplanes over finite fields behave according to standard linear algebra
Invoked when translating rank conditions into intersection conditions with hyperplanes.

invented entities (2)

systems associated to a code no independent evidence
purpose: Translate metric properties into geometric intersection conditions
Core new object defined from generator tensors and slice spaces.
evasive systems no independent evidence
purpose: Define generalized weights for the codes
Introduced as an application to capture weight distributions geometrically.

pith-pipeline@v0.9.0 · 5686 in / 1299 out tokens · 41434 ms · 2026-05-20T04:32:13.349439+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

To every nondegenerate matrix rank-metric code, we associate two systems, which translate metric properties of the code into geometric conditions involving intersections with hyperplanes.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

This leads to a correspondence between equivalence classes of nondegenerate matrix rank-metric codes and equivalence classes of systems

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

37 extracted references · 37 canonical work pages · 3 internal anchors

[1]

G. N. Alfarano, M. Borello, A. Neri, and A. Ravagnani. Linear cutting blocking sets and minimal codes in the rank metric.J. Comb. Theory Ser. A, 192:105658, 2022

work page 2022
[2]

G. N. Alfarano and E. Byrne. Recursive properties of the characteristic polynomial of weighted lattices.Adv. Appl. Math., 176:103046, 2026

work page 2026
[3]

G. N. Alfarano and S. Degen. Representability ofq-matroids via rank-metric codes.arXiv preprint arXiv:2605.15780, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[4]

S. Ball, G. Gamboa, and M. Lavrauw. On additive MDS codes over small fields.Adv. Math. Commun., 17(4):828– 844, 2023

work page 2023
[5]

S. Ball, M. Lavrauw, and T. Popatia. Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes.Des. Codes Cryptogr., 93(1):175–196, 2025

work page 2025
[6]

L. B. Beasley and T. J. Laffey. Linear operators on matrices: the invariance of rank-k matrices.Linear Algebra Appl., 133:175–184, 1990

work page 1990
[7]

Bonisoli

A. Bonisoli. Every equidistant linear code is a sequence of dual Hamming codes.Ars Comb., 18:181–186, 1984

work page 1984
[8]

Boralevi, D

A. Boralevi, D. Faenzi, and E. Mezzetti. Linear spaces of matrices of constant rank and instanton bundles.Adv. Math., 248:895–920, 2013. THE GEOMETRY OF RANK-METRIC CODES 31

work page 2013
[9]

N. Boston. Spaces of constant rank matrices over GF(2).Electron. J. Linear Algebra, 20:1–5, 2010

work page 2010
[10]

Britz, A

T. Britz, A. Mammoliti, and K. Shiromoto. Wei-type duality theorems for rank metric codes.Des. Codes Cryp- togr., 88(8):1503–1519, 2020

work page 2020
[11]

Byrne, A

E. Byrne, A. Neri, A. Ravagnani, and J. Sheekey. Tensor representation of rank-metric codes.SIAM J. Appl. Algebra Geom., 3(4):614–643, 2019

work page 2019
[12]

Byrne and A

E. Byrne and A. Ravagnani. Covering radius of matrix codes endowed with the rank metric.SIAM J. Discrete Math., 31(2):927–944, 2017

work page 2017
[13]

Byrne and J

E. Byrne and J. Sheekey. Bounds on the critical exponent of matrix codes through tensor representations.preprint, 2026

work page 2026
[14]

Cooperstein.Advanced linear algebra, volume 2

B. Cooperstein.Advanced linear algebra, volume 2. CRC Press, 2010

work page 2010
[15]

Delsarte

P. Delsarte. Bilinear forms over a finite field, with applications to coding theory.J. Comb. Theory Ser. A, 25(3):226–241, 1978

work page 1978
[16]

Generalized Hamming weights of additive codes and geometric counterparts

J. D’haeseleer and S. Kurz. Generalized Hamming weights of additive codes and geometric counterparts.arXiv preprint arXiv:2512.16327, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[17]

Dumas, R

J.-G. Dumas, R. Gow, G. McGuire, and J. Sheekey. Subspaces of matrices with special rank properties.Linear Algebra Appl., 433(1):191–202, 2010

work page 2010
[18]

E. M. Gabidulin. Theory of codes with maximum rank distance.Problemy Peredachi Informatsii, 21(1):3–16, 1985

work page 1985
[19]

S. R. Ghorpade and T. Johnsen. A polymatroid approach to generalized weights of rank metric codes.Des. Codes Cryptogr., 88(12):2531–2546, 2020

work page 2020
[20]

E. Gorla. Rank-metric codes. InConcise Encyclopedia of Coding Theory, pages 227–250. Chapman and Hall/CRC, 2021

work page 2021
[21]

Helleseth, T

T. Helleseth, T. Kløve, and J. Mykkeltveit. The weight distribution of irreducible cyclic codes with block lengths n1((qℓ −1)/n).Discrete Math., 18(2):179–211, 1977

work page 1977
[22]

D. E. Knuth. Finite semifields and projective planes.J. Algebra, 2(2):182–217, 1965

work page 1965
[23]

S. Kurz. Additive codes attaining the Griesmer bound.arXiv preprint arXiv:2412.14615, 2024

work page internal anchor Pith review Pith/arXiv arXiv 2024
[24]

Landsberg and L

J. Landsberg and L. Manivel. Equivariant spaces of matrices of constant rank.Pure Appl. Math. Q., 21(3):1209– 1244, 2024

work page 2024
[25]

Lunardon

G. Lunardon. Normal spreads.Geometriae Dedicata, 75(3):245–261, 1999

work page 1999
[26]

Lunardon, R

G. Lunardon, R. Trombetti, and Y. Zhou. Generalized twisted Gabidulin codes.J. Comb. Theory Ser. A, 159:79– 106, 2018

work page 2018
[27]

Marino, A

G. Marino, A. Neri, and R. Trombetti. Evasive subspaces, generalized rank weights and near MRD codes.Discrete Math., 346(12):113605, 2023

work page 2023
[28]

Mart´ ınez-Pe˜ nas and R

U. Mart´ ınez-Pe˜ nas and R. Matsumoto. Relative generalized matrix weights of matrix codes for universal security on wire-tap networks.IEEE Trans. Inf. Theory, 64(4):2529–2549, 2017

work page 2017
[29]

A. Neri, P. Santonastaso, and F. Zullo. The geometry of one-weight codes in the sum-rank metric.J. Comb. Theory Ser. A, 194:105703, 2023

work page 2023
[30]

Oggier and A

F. Oggier and A. Sboui. On the existence of generalized rank weights. In2012 International Symposium on Information Theory and its Applications, pages 406–410. IEEE, 2012

work page 2012
[31]

Polverino

O. Polverino. Linear sets in finite projective spaces.Discrete Math., 310(22):3096–3107, 2010

work page 2010
[32]

T. H. Randrianarisoa. A geometric approach to rank metric codes and a classification of constant weight codes. Des. Codes Cryptogr., 88(7):1331–1348, 2020

work page 2020
[33]

Ravagnani

A. Ravagnani. Generalized weights: An anticode approach.J. Pure Appl. Algebra, 220(5):1946–1962, 2016

work page 1946
[34]

B. Segre. Curve razionali normali ek-archi negli spazi finiti.Ann. Mat. Pura Appl., 39(1):357–379, 1955

work page 1955
[35]

J. Sheekey. (Scattered) linear sets are to rank-metric codes as arcs are to Hamming-metric codes. In M. Greferath, C. Hollanti, and J. Rosenthal, editors,Oberwolfach Report No. 13/2019, 2019

work page 2019
[36]

Tsfasman and S

M. Tsfasman and S. G. Vladut.Algebraic-geometric codes, volume 58. Springer Science & Business Media, 2013

work page 2013
[37]

V. K. Wei. Generalized Hamming weights for linear codes.IEEE Trans. Inf. Theory, 37(5):1412–1418, 1991

work page 1991

[1] [1]

G. N. Alfarano, M. Borello, A. Neri, and A. Ravagnani. Linear cutting blocking sets and minimal codes in the rank metric.J. Comb. Theory Ser. A, 192:105658, 2022

work page 2022

[2] [2]

G. N. Alfarano and E. Byrne. Recursive properties of the characteristic polynomial of weighted lattices.Adv. Appl. Math., 176:103046, 2026

work page 2026

[3] [3]

G. N. Alfarano and S. Degen. Representability ofq-matroids via rank-metric codes.arXiv preprint arXiv:2605.15780, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[4] [4]

S. Ball, G. Gamboa, and M. Lavrauw. On additive MDS codes over small fields.Adv. Math. Commun., 17(4):828– 844, 2023

work page 2023

[5] [5]

S. Ball, M. Lavrauw, and T. Popatia. Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes.Des. Codes Cryptogr., 93(1):175–196, 2025

work page 2025

[6] [6]

L. B. Beasley and T. J. Laffey. Linear operators on matrices: the invariance of rank-k matrices.Linear Algebra Appl., 133:175–184, 1990

work page 1990

[7] [7]

Bonisoli

A. Bonisoli. Every equidistant linear code is a sequence of dual Hamming codes.Ars Comb., 18:181–186, 1984

work page 1984

[8] [8]

Boralevi, D

A. Boralevi, D. Faenzi, and E. Mezzetti. Linear spaces of matrices of constant rank and instanton bundles.Adv. Math., 248:895–920, 2013. THE GEOMETRY OF RANK-METRIC CODES 31

work page 2013

[9] [9]

N. Boston. Spaces of constant rank matrices over GF(2).Electron. J. Linear Algebra, 20:1–5, 2010

work page 2010

[10] [10]

Britz, A

T. Britz, A. Mammoliti, and K. Shiromoto. Wei-type duality theorems for rank metric codes.Des. Codes Cryp- togr., 88(8):1503–1519, 2020

work page 2020

[11] [11]

Byrne, A

E. Byrne, A. Neri, A. Ravagnani, and J. Sheekey. Tensor representation of rank-metric codes.SIAM J. Appl. Algebra Geom., 3(4):614–643, 2019

work page 2019

[12] [12]

Byrne and A

E. Byrne and A. Ravagnani. Covering radius of matrix codes endowed with the rank metric.SIAM J. Discrete Math., 31(2):927–944, 2017

work page 2017

[13] [13]

Byrne and J

E. Byrne and J. Sheekey. Bounds on the critical exponent of matrix codes through tensor representations.preprint, 2026

work page 2026

[14] [14]

Cooperstein.Advanced linear algebra, volume 2

B. Cooperstein.Advanced linear algebra, volume 2. CRC Press, 2010

work page 2010

[15] [15]

Delsarte

P. Delsarte. Bilinear forms over a finite field, with applications to coding theory.J. Comb. Theory Ser. A, 25(3):226–241, 1978

work page 1978

[16] [16]

Generalized Hamming weights of additive codes and geometric counterparts

J. D’haeseleer and S. Kurz. Generalized Hamming weights of additive codes and geometric counterparts.arXiv preprint arXiv:2512.16327, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[17] [17]

Dumas, R

J.-G. Dumas, R. Gow, G. McGuire, and J. Sheekey. Subspaces of matrices with special rank properties.Linear Algebra Appl., 433(1):191–202, 2010

work page 2010

[18] [18]

E. M. Gabidulin. Theory of codes with maximum rank distance.Problemy Peredachi Informatsii, 21(1):3–16, 1985

work page 1985

[19] [19]

S. R. Ghorpade and T. Johnsen. A polymatroid approach to generalized weights of rank metric codes.Des. Codes Cryptogr., 88(12):2531–2546, 2020

work page 2020

[20] [20]

E. Gorla. Rank-metric codes. InConcise Encyclopedia of Coding Theory, pages 227–250. Chapman and Hall/CRC, 2021

work page 2021

[21] [21]

Helleseth, T

T. Helleseth, T. Kløve, and J. Mykkeltveit. The weight distribution of irreducible cyclic codes with block lengths n1((qℓ −1)/n).Discrete Math., 18(2):179–211, 1977

work page 1977

[22] [22]

D. E. Knuth. Finite semifields and projective planes.J. Algebra, 2(2):182–217, 1965

work page 1965

[23] [23]

S. Kurz. Additive codes attaining the Griesmer bound.arXiv preprint arXiv:2412.14615, 2024

work page internal anchor Pith review Pith/arXiv arXiv 2024

[24] [24]

Landsberg and L

J. Landsberg and L. Manivel. Equivariant spaces of matrices of constant rank.Pure Appl. Math. Q., 21(3):1209– 1244, 2024

work page 2024

[25] [25]

Lunardon

G. Lunardon. Normal spreads.Geometriae Dedicata, 75(3):245–261, 1999

work page 1999

[26] [26]

Lunardon, R

G. Lunardon, R. Trombetti, and Y. Zhou. Generalized twisted Gabidulin codes.J. Comb. Theory Ser. A, 159:79– 106, 2018

work page 2018

[27] [27]

Marino, A

G. Marino, A. Neri, and R. Trombetti. Evasive subspaces, generalized rank weights and near MRD codes.Discrete Math., 346(12):113605, 2023

work page 2023

[28] [28]

Mart´ ınez-Pe˜ nas and R

U. Mart´ ınez-Pe˜ nas and R. Matsumoto. Relative generalized matrix weights of matrix codes for universal security on wire-tap networks.IEEE Trans. Inf. Theory, 64(4):2529–2549, 2017

work page 2017

[29] [29]

A. Neri, P. Santonastaso, and F. Zullo. The geometry of one-weight codes in the sum-rank metric.J. Comb. Theory Ser. A, 194:105703, 2023

work page 2023

[30] [30]

Oggier and A

F. Oggier and A. Sboui. On the existence of generalized rank weights. In2012 International Symposium on Information Theory and its Applications, pages 406–410. IEEE, 2012

work page 2012

[31] [31]

Polverino

O. Polverino. Linear sets in finite projective spaces.Discrete Math., 310(22):3096–3107, 2010

work page 2010

[32] [32]

T. H. Randrianarisoa. A geometric approach to rank metric codes and a classification of constant weight codes. Des. Codes Cryptogr., 88(7):1331–1348, 2020

work page 2020

[33] [33]

Ravagnani

A. Ravagnani. Generalized weights: An anticode approach.J. Pure Appl. Algebra, 220(5):1946–1962, 2016

work page 1946

[34] [34]

B. Segre. Curve razionali normali ek-archi negli spazi finiti.Ann. Mat. Pura Appl., 39(1):357–379, 1955

work page 1955

[35] [35]

J. Sheekey. (Scattered) linear sets are to rank-metric codes as arcs are to Hamming-metric codes. In M. Greferath, C. Hollanti, and J. Rosenthal, editors,Oberwolfach Report No. 13/2019, 2019

work page 2019

[36] [36]

Tsfasman and S

M. Tsfasman and S. G. Vladut.Algebraic-geometric codes, volume 58. Springer Science & Business Media, 2013

work page 2013

[37] [37]

V. K. Wei. Generalized Hamming weights for linear codes.IEEE Trans. Inf. Theory, 37(5):1412–1418, 1991

work page 1991