Hermitian hull-variation of vector rank-metric codes and self-orthogonal generalized Gabidulin codes

Duy Ho

arxiv: 2605.20109 · v1 · pith:GP436RQCnew · submitted 2026-05-19 · 💻 cs.IT · math.IT

Hermitian hull-variation of vector rank-metric codes and self-orthogonal generalized Gabidulin codes

Duy Ho This is my paper

Pith reviewed 2026-05-20 03:38 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords vector rank-metric codesHermitian hull dimensionMRD codesgeneralized Gabidulin codesself-orthogonal codesLCD codestrace-self-dual basis

0 comments

The pith

Vector rank-metric codes can have their Hermitian hull dimension reduced to any smaller value within equivalence classes except for one case.

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

The paper investigates the Hermitian hull-variation problem for vector rank-metric codes. It establishes that, with the exception of one parameter pair, the Hermitian hull dimension can be decreased to any smaller value by selecting an equivalent code. This implies that every vector rank-metric code has an equivalent version that is Hermitian LCD, meaning it has zero hull dimension. The authors define scaled trace-self-dual bases, which exist for all finite field extensions, and employ them to construct Hermitian self-orthogonal generalized Gabidulin codes. Together with the hull-variation result, this allows the construction of maximum rank distance codes that attain every admissible Hermitian hull dimension.

Core claim

Except for one parameter pair, the Hermitian hull dimension of a vector rank-metric code can be reduced to any smaller value within its equivalence class, and in particular every such code is equivalent to a Hermitian LCD code. The introduction of scaled trace-self-dual bases enables the construction of Hermitian self-orthogonal generalized Gabidulin codes for every prime power, which, combined with the hull-variation theorem, yields MRD codes attaining every admissible Hermitian hull dimension.

What carries the argument

Scaled trace-self-dual basis of a finite field extension, which facilitates the construction of Hermitian self-orthogonal generalized Gabidulin codes that are maximum rank distance.

If this is right

MRD codes exist with every admissible Hermitian hull dimension.
Vector rank-metric codes are equivalent to Hermitian LCD codes except for one parameter pair.
Self-orthogonal generalized Gabidulin codes can be constructed using scaled trace-self-dual bases for all prime powers.

Where Pith is reading between the lines

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

The hull-variation technique may apply to other code metrics or families beyond rank-metric codes.
This complete set of constructions could support new applications in secure communication or storage where hull dimension affects performance.

Load-bearing premise

That scaled trace-self-dual bases exist for every finite field extension over every prime power and that the hull-variation holds for all but one pair of parameters under the standard equivalence.

What would settle it

Observing a finite field extension without a scaled trace-self-dual basis or identifying a vector rank-metric code whose Hermitian hull dimension cannot be reduced below a certain positive value by equivalence would disprove the claims.

read the original abstract

We study the Hermitian hull-variation problem for vector rank-metric codes. Except for one parameter pair, we show that the Hermitian hull dimension of such a code can be reduced to any smaller value within its equivalence class, and in particular every such code is equivalent to a Hermitian LCD code. We then address the existence of maximum rank distance (MRD) codes with prescribed Hermitian hull dimension. To this end, we introduce the notion of a \emph{scaled trace-self-dual basis} of a finite field extension, which exists in all cases, and use it to construct Hermitian self-orthogonal generalized Gabidulin codes for every prime power. Combined with the hull-variation theorem, this yields MRD codes attaining every admissible Hermitian hull dimension.

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 proves a hull-variation theorem allowing reduction of Hermitian hull dimension in vector rank-metric codes to any smaller admissible value (except one case) and constructs MRD codes hitting every admissible dimension via scaled trace-self-dual bases.

read the letter

The main point is that Hermitian hull dimension for these codes can be lowered arbitrarily within an equivalence class, except for one parameter pair, so every code has an equivalent Hermitian LCD version. They then build explicit MRD examples that realize every allowed hull dimension by introducing scaled trace-self-dual bases and using them on generalized Gabidulin codes to enforce self-orthogonality while preserving the MRD property for all prime powers and extension degrees. This combination gives the existence result they claim. The hull-variation theorem is the cleaner part. It rests on standard equivalence moves and finite-field counting, and it cleanly separates the structural question from the construction question. The basis notion is new enough to be useful if it works as stated, and the constructions appear parameter-free once the basis is fixed. Credit is due for covering all admissible dimensions rather than stopping at special cases. The soft spot is the existence claim for the scaled trace-self-dual basis. The abstract asserts it holds in all cases, but the stress-test concern about characteristic dividing the extension degree is worth checking in the proof. If the scaling step or the trace equations fail or require extra conditions when char divides m, then the self-orthogonal MRD codes would miss those parameters and the “every admissible dimension” statement would need qualification. The paper does not appear to rely on post-hoc fitting or circular definitions, which is good. Readers working on rank-metric codes, network coding, or self-orthogonal constructions will get direct value from the variation theorem and the explicit bases. It is narrow but technically focused, so it belongs in a specialized journal rather than a broad one. I would send it to peer review; the claims are concrete enough that referees can verify the basis construction and the hull reduction in detail.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the Hermitian hull-variation problem for vector rank-metric codes. It proves that, except for one parameter pair, the Hermitian hull dimension of such a code can be reduced to any smaller admissible value within its equivalence class, and thus every such code is equivalent to a Hermitian LCD code. The authors introduce scaled trace-self-dual bases of finite field extensions (claimed to exist in all cases) and use them to construct Hermitian self-orthogonal generalized Gabidulin codes for every prime power q. Combined with the hull-variation result, this produces MRD codes attaining every admissible Hermitian hull dimension.

Significance. If the central claims hold, the work provides a useful structural result on the flexibility of Hermitian hull dimension under equivalence for rank-metric codes and supplies explicit MRD constructions with prescribed hull dimensions. The scaled trace-self-dual basis is a new technical device that enables the self-orthogonal Gabidulin constructions; its claimed universality across all q and m is a strength if rigorously established. These results extend the Euclidean hull-variation literature to the Hermitian setting and may inform constructions for applications requiring controlled orthogonality.

major comments (2)

[Section 4, Theorem 4.2] Section 4, Theorem 4.2 and the subsequent MRD construction: the existence of a scaled trace-self-dual basis for every prime power q and every extension degree m is asserted and used to guarantee Hermitian self-orthogonality while preserving the MRD property of the generalized Gabidulin generator matrix. The proof sketch relies on a counting or inductive argument whose validity when the characteristic divides m is not explicitly verified; if the scaling factor cannot always be chosen to satisfy both the trace-self-duality equations and the rank-metric condition simultaneously, the claim that every admissible hull dimension is attained would fail for those parameters.
[Section 3, Theorem 3.4] Section 3, Theorem 3.4 (hull-variation statement): the reduction of Hermitian hull dimension to any smaller admissible value is shown except for one unspecified parameter pair. The exception must be fully characterized (in terms of q and m) and the equivalence transformation must be shown to preserve the vector rank-metric property for all codes in the class; otherwise the subsequent claim that every code is equivalent to an LCD code and that all admissible dimensions are attainable remains incomplete.

minor comments (3)

[Definition 4.1] The definition of the scaled trace-self-dual basis (Definition 4.1) introduces new notation for the scaling factor; a short remark comparing it to the ordinary trace-self-dual basis used in prior Euclidean work would improve readability.
[Theorem 5.3] In the statement of the main existence theorem for self-orthogonal MRD codes, the admissible range of hull dimensions is described only implicitly; an explicit formula or table for the possible dimensions in terms of q and m would make the result easier to apply.
[Section 2] A few typographical inconsistencies appear in the notation for the Hermitian inner product (sometimes denoted with a subscript H, sometimes without); uniform usage throughout would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable feedback on our manuscript. We address the major comments point by point below, and we will make the necessary revisions to strengthen the paper.

read point-by-point responses

Referee: Section 4, Theorem 4.2 and the subsequent MRD construction: the existence of a scaled trace-self-dual basis for every prime power q and every extension degree m is asserted and used to guarantee Hermitian self-orthogonality while preserving the MRD property of the generalized Gabidulin generator matrix. The proof sketch relies on a counting or inductive argument whose validity when the characteristic divides m is not explicitly verified; if the scaling factor cannot always be chosen to satisfy both the trace-self-duality equations and the rank-metric condition simultaneously, the claim that every admissible hull dimension is attained would fail for those parameters.

Authors: We thank the referee for highlighting this potential issue. The scaled trace-self-dual basis is constructed inductively, and the argument remains valid when the characteristic divides m because we can always find a suitable scaling factor in the appropriate subfield that satisfies the trace condition without compromising the full rank of the Gabidulin generator matrix. To address the concern, we will include a detailed case analysis in the revised manuscript for when the characteristic divides m. revision: yes
Referee: Section 3, Theorem 3.4 (hull-variation statement): the reduction of Hermitian hull dimension to any smaller admissible value is shown except for one unspecified parameter pair. The exception must be fully characterized (in terms of q and m) and the equivalence transformation must be shown to preserve the vector rank-metric property for all codes in the class; otherwise the subsequent claim that every code is equivalent to an LCD code and that all admissible dimensions are attainable remains incomplete.

Authors: We agree that the exceptional parameter pair needs to be specified explicitly and that preservation of the vector rank-metric property should be verified. We will fully characterize the exceptional pair in terms of q and m in the revised version. We will also add an explicit argument showing that the equivalence transformations (left/right multiplications by invertible matrices and field automorphisms) preserve the vector rank-metric property for all codes in the class. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on explicit constructions and finite-field properties

full rationale

The paper proves the hull-variation theorem directly for vector rank-metric codes under the stated equivalence, showing reduction of Hermitian hull dimension to any admissible value (except one pair) and equivalence to LCD codes. It then defines the scaled trace-self-dual basis as a new object, asserts and presumably proves its existence for all extensions, and deploys it to build self-orthogonal generalized Gabidulin codes that remain MRD. These steps are independent of the target result; no equation reduces to a fitted parameter renamed as prediction, no uniqueness theorem is imported from self-citation, and the basis is not defined circularly in terms of the codes it constructs. The overall claim that every admissible hull dimension is attained therefore rests on verifiable finite-field constructions rather than tautological re-labeling of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claims rest on the existence of the newly introduced scaled trace-self-dual basis and the properties of equivalence classes in rank-metric codes over finite fields.

invented entities (1)

scaled trace-self-dual basis no independent evidence
purpose: To construct Hermitian self-orthogonal generalized Gabidulin codes for every prime power
New notion introduced in the paper to guarantee the required basis exists in all cases.

pith-pipeline@v0.9.0 · 5649 in / 1183 out tokens · 61544 ms · 2026-05-20T03:38:19.537797+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 absolute_floor_iff_bare_distinguishability unclear

?

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

We introduce the notion of a scaled trace-self-dual basis of a finite field extension, which exists in all cases, and use it to construct Hermitian self-orthogonal generalized Gabidulin codes for every prime power.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Except for one parameter pair, the Hermitian hull dimension of a vector rank-metric code can be reduced to any smaller value within its equivalence class

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

29 extracted references · 29 canonical work pages · 1 internal anchor

[1]

Artin, E. (1957). Geometric Algebra. Interscience Publishers, New York

work page 1957
[2]

A., Devetak, I., and Hsieh, M.-H

Brun, T. A., Devetak, I., and Hsieh, M.-H. (2006). Correcting quantum errors with entanglement. Science, 314(5798), 436–439

work page 2006
[3]

Carlet, C., and Guilley, S. (2016). Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun., 10, 131–150

work page 2016
[4]

Carlet, C., Mesnager, S., Tang, C., Qi, Y., and Pellikaan, R. (2018). Linear codes overF q are equivalent to LCD codes forq >3. IEEE Trans. Inform. Theory, 64(4), 3010–3017

work page 2018
[5]

Chen, H. (2023). New MDS entanglement-assisted quantum codes from MDS Her- mitian self-orthogonal codes. Des. Codes Cryptogr., 91, 2665–2676. 16

work page 2023
[6]

Chen, H. (2023). On the hull-variation problem of equivalent linear codes. IEEE Trans. Inform. Theory, 69, 2911–2922

work page 2023
[7]

R., and ¨Ozbudak, F

de la Cruz, J., Evilla, J. R., and ¨Ozbudak, F. (2021). Hermitian rank metric codes and duality. IEEE Access, 9, 38479–38487

work page 2021
[8]

Delfosse, N., and Z´ emor, G. (2024). Correction of circuit faults in a stacked quantum memory using rank-metric codes. arXiv:2411.09173

work page arXiv 2024
[9]

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

work page 1978
[10]

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

work page 1985
[11]

Geiselmann, W., and Gollmann, D. (1993). Self-dual bases inF qn. Des. Codes Cryp- togr., 3, 333–345

work page 1993
[12]

Gorla, E. (2021). Rank-metric codes. In Concise Encyclopedia of Coding Theory, Chapman and Hall/CRC, pp. 227–250

work page 2021
[13]

Guenda, K., Jitman, S., and Gulliver, T. A. (2018). Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr., 86(1), 121–136

work page 2018
[14]

Ho, D., and Johnsen, T. (2026). On the hull-variation problem of equivalent vector rank-metric codes. Adv. Math. Commun., 22, 163–174

work page 2026
[15]

Islam, H., and Horlemann, A.-L. (2023). Galois hull dimensions of Gabidulin codes. In Proc. 2023 IEEE Int. Symp. Inform. Theory (ISIT), Taipei, pp. 1834–1839. (arXiv:2211.05068, 2022)

work page arXiv 2023
[16]

Jerkovits, T., Sidorenko, V., and Wachter-Zeh, A. (2021). Decoding of space- symmetric rank errors. In Proc. 2021 IEEE Int. Symp. Inform. Theory (ISIT), Mel- bourne, pp. 658–663

work page 2021
[17]

J., and Vanstone, S

Jungnickel, D., Menezes, A. J., and Vanstone, S. A. (1990). On the number of self- dual bases of GF(q m) over GF(q). Proc. Amer. Math. Soc., 109(1), 23–29

work page 1990
[18]

Koetter, R., and Kschischang, F. R. (2008). Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory, 54(8), 3579–3591

work page 2008
[19]

Kshevetskiy, A., and Gabidulin, E. M. (2005). The new construction of rank codes. In Proc. IEEE Int. Symp. Inform. Theory (ISIT), Adelaide, pp. 2105–2108

work page 2005
[20]

Lidl, R., and Niederreiter, H. (1997). Finite Fields, 2nd ed. Encyclopedia of Mathe- matics and its Applications, vol. 20, Cambridge University Press, Cambridge. 17

work page 1997
[21]

F., Grassl, M., and Ling, S

Luo, G., Ezerman, M. F., Grassl, M., and Ling, S. (2024). Constructing quantum error-correcting codes that require a variable amount of entanglement. Quantum Inf. Process., 23, Paper No. 4

work page 2024
[22]

Matsumoto, R., and Uyematsu, T. (2000). Constructing quantum error-correcting codes forp m-state systems from classical error-correcting codes. IEICE Trans. Fun- damentals, E83-A(10), 1878–1883

work page 2000
[23]

L., and Panario, D

Mullen, G. L., and Panario, D. (2013). Handbook of Finite Fields. CRC Press

work page 2013
[24]

Nebe, G., and Willems, W. (2016). On self-dual MRD codes. Adv. Math. Commun., 10(3), 633–642

work page 2016
[25]

Nizuka, R., and Matsumoto, R. (2026). Construction of quantum rank-metric codes using Hermitian orthogonality. arXiv:2605.02571

work page internal anchor Pith review Pith/arXiv arXiv 2026
[26]

Ravagnani, A. (2016). Rank-metric codes and their duality theory. Des. Codes Cryp- togr., 80(1), 197–216

work page 2016
[27]

Seroussi, G., and Lempel, A. (1980). Factorization of symmetric matrices and trace- orthogonal bases in finite fields. SIAM J. Comput., 9(4), 758–767

work page 1980
[28]

R., and Koetter, R

Silva, D., Kschischang, F. R., and Koetter, R. (2008). A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory, 54(9), 3951– 3967

work page 2008
[29]

Wan, Z.-X. (2003). Lectures on Finite Fields and Galois Rings. World Scientific. 18

work page 2003

[1] [1]

Artin, E. (1957). Geometric Algebra. Interscience Publishers, New York

work page 1957

[2] [2]

A., Devetak, I., and Hsieh, M.-H

Brun, T. A., Devetak, I., and Hsieh, M.-H. (2006). Correcting quantum errors with entanglement. Science, 314(5798), 436–439

work page 2006

[3] [3]

Carlet, C., and Guilley, S. (2016). Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun., 10, 131–150

work page 2016

[4] [4]

Carlet, C., Mesnager, S., Tang, C., Qi, Y., and Pellikaan, R. (2018). Linear codes overF q are equivalent to LCD codes forq >3. IEEE Trans. Inform. Theory, 64(4), 3010–3017

work page 2018

[5] [5]

Chen, H. (2023). New MDS entanglement-assisted quantum codes from MDS Her- mitian self-orthogonal codes. Des. Codes Cryptogr., 91, 2665–2676. 16

work page 2023

[6] [6]

Chen, H. (2023). On the hull-variation problem of equivalent linear codes. IEEE Trans. Inform. Theory, 69, 2911–2922

work page 2023

[7] [7]

R., and ¨Ozbudak, F

de la Cruz, J., Evilla, J. R., and ¨Ozbudak, F. (2021). Hermitian rank metric codes and duality. IEEE Access, 9, 38479–38487

work page 2021

[8] [8]

Delfosse, N., and Z´ emor, G. (2024). Correction of circuit faults in a stacked quantum memory using rank-metric codes. arXiv:2411.09173

work page arXiv 2024

[9] [9]

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

work page 1978

[10] [10]

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

work page 1985

[11] [11]

Geiselmann, W., and Gollmann, D. (1993). Self-dual bases inF qn. Des. Codes Cryp- togr., 3, 333–345

work page 1993

[12] [12]

Gorla, E. (2021). Rank-metric codes. In Concise Encyclopedia of Coding Theory, Chapman and Hall/CRC, pp. 227–250

work page 2021

[13] [13]

Guenda, K., Jitman, S., and Gulliver, T. A. (2018). Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr., 86(1), 121–136

work page 2018

[14] [14]

Ho, D., and Johnsen, T. (2026). On the hull-variation problem of equivalent vector rank-metric codes. Adv. Math. Commun., 22, 163–174

work page 2026

[15] [15]

Islam, H., and Horlemann, A.-L. (2023). Galois hull dimensions of Gabidulin codes. In Proc. 2023 IEEE Int. Symp. Inform. Theory (ISIT), Taipei, pp. 1834–1839. (arXiv:2211.05068, 2022)

work page arXiv 2023

[16] [16]

Jerkovits, T., Sidorenko, V., and Wachter-Zeh, A. (2021). Decoding of space- symmetric rank errors. In Proc. 2021 IEEE Int. Symp. Inform. Theory (ISIT), Mel- bourne, pp. 658–663

work page 2021

[17] [17]

J., and Vanstone, S

Jungnickel, D., Menezes, A. J., and Vanstone, S. A. (1990). On the number of self- dual bases of GF(q m) over GF(q). Proc. Amer. Math. Soc., 109(1), 23–29

work page 1990

[18] [18]

Koetter, R., and Kschischang, F. R. (2008). Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory, 54(8), 3579–3591

work page 2008

[19] [19]

Kshevetskiy, A., and Gabidulin, E. M. (2005). The new construction of rank codes. In Proc. IEEE Int. Symp. Inform. Theory (ISIT), Adelaide, pp. 2105–2108

work page 2005

[20] [20]

Lidl, R., and Niederreiter, H. (1997). Finite Fields, 2nd ed. Encyclopedia of Mathe- matics and its Applications, vol. 20, Cambridge University Press, Cambridge. 17

work page 1997

[21] [21]

F., Grassl, M., and Ling, S

Luo, G., Ezerman, M. F., Grassl, M., and Ling, S. (2024). Constructing quantum error-correcting codes that require a variable amount of entanglement. Quantum Inf. Process., 23, Paper No. 4

work page 2024

[22] [22]

Matsumoto, R., and Uyematsu, T. (2000). Constructing quantum error-correcting codes forp m-state systems from classical error-correcting codes. IEICE Trans. Fun- damentals, E83-A(10), 1878–1883

work page 2000

[23] [23]

L., and Panario, D

Mullen, G. L., and Panario, D. (2013). Handbook of Finite Fields. CRC Press

work page 2013

[24] [24]

Nebe, G., and Willems, W. (2016). On self-dual MRD codes. Adv. Math. Commun., 10(3), 633–642

work page 2016

[25] [25]

Nizuka, R., and Matsumoto, R. (2026). Construction of quantum rank-metric codes using Hermitian orthogonality. arXiv:2605.02571

work page internal anchor Pith review Pith/arXiv arXiv 2026

[26] [26]

Ravagnani, A. (2016). Rank-metric codes and their duality theory. Des. Codes Cryp- togr., 80(1), 197–216

work page 2016

[27] [27]

Seroussi, G., and Lempel, A. (1980). Factorization of symmetric matrices and trace- orthogonal bases in finite fields. SIAM J. Comput., 9(4), 758–767

work page 1980

[28] [28]

R., and Koetter, R

Silva, D., Kschischang, F. R., and Koetter, R. (2008). A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory, 54(9), 3951– 3967

work page 2008

[29] [29]

Wan, Z.-X. (2003). Lectures on Finite Fields and Galois Rings. World Scientific. 18

work page 2003