Constructions of Rank-Metric Codes of Small Tensor Rank

Eimear Byrne; Giuseppe Cotardo; Matteo Bonini

arxiv: 2605.21784 · v1 · pith:2ZBC27GUnew · submitted 2026-05-20 · 💻 cs.IT · math.AG· math.CO· math.IT

Constructions of Rank-Metric Codes of Small Tensor Rank

Matteo Bonini , Eimear Byrne , Giuseppe Cotardo This is my paper

Pith reviewed 2026-05-22 07:37 UTC · model grok-4.3

classification 💻 cs.IT math.AGmath.COmath.IT

keywords rank-metric codestensor ranktensor rank defectalgebraic geometry codesMDS codesHamming metricfinite fieldsminimum distance

0 comments

The pith

Rank-metric codes achieve small tensor rank defect when built from algebraic geometry codes via Hamming-metric parameters.

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

The paper establishes a direct link between the tensor rank of a rank-metric code and the dimension and minimum distance of associated linear codes measured in the Hamming metric. It introduces tensor rank defect as the gap between a code's actual tensor rank and the lower bound of k plus d minus one. Algebraic geometry codes are then used to produce explicit rank-metric codes whose defect is small. This matters because minimal-tensor-rank codes are known to imply the existence of maximum distance separable codes, which achieve optimal performance in error correction over finite fields. The constructions therefore enlarge the set of known near-optimal rank-metric codes.

Core claim

The authors prove that tensor rank of a rank-metric code is bounded in terms of the parameters of its associated Hamming-metric code, define tensor rank defect as the excess over the Kruskal bound, and construct families of rank-metric codes with small defect by starting from algebraic geometry codes.

What carries the argument

The mapping from a rank-metric code to an associated linear code in the Hamming metric, which supplies bounds on tensor rank and allows the tensor rank defect to be controlled through algebraic geometry code parameters.

If this is right

New explicit rank-metric codes exist whose tensor rank lies close to the Kruskal lower bound of k + d - 1.
The defect can be made arbitrarily small in certain parameter regimes by choosing algebraic geometry codes with sufficiently large minimum distance.
Each such construction yields a corresponding maximum distance separable code via the known implication from minimal tensor rank codes.
The relation between the two metrics supplies an explicit way to compute or estimate tensor rank without enumerating all possible decompositions.

Where Pith is reading between the lines

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

The same association might be used to study tensor rank in other families such as Reed-Muller or cyclic codes.
Small-defect codes could reduce computational cost in applications like distributed matrix multiplication that rely on low-rank decompositions.
If the defect can be driven to zero for new parameters, the constructions would produce previously unknown minimal tensor rank codes and therefore new MDS codes.

Load-bearing premise

The minimum distance and dimension of the associated Hamming-metric code can be used to bound the tensor rank of the rank-metric code, and algebraic geometry codes provide parameters that keep this bound tight enough for small defect.

What would settle it

Direct calculation of tensor rank for a concrete small-length rank-metric code obtained from a known algebraic geometry code, followed by comparison against the defect value predicted from the Hamming-metric parameters of the associated code.

read the original abstract

Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one tensors. Kruskal showed that the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k + d - 1$, and codes meeting this bound with equality are called minimal tensor rank (MTR) codes. It is known from algebraic complexity theory that the existence of an MTR code implies the existence of a maximum distance separable (MDS) code. In this work, we establish new results relating the tensor rank of a rank-metric code to the parameters of associated linear codes in the Hamming metric and introduce the notion of tensor rank defect. We then develop new constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes.

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 defines tensor rank defect for rank-metric codes and gives AG-code constructions that achieve small defect beyond the Kruskal bound.

read the letter

The core contribution is the definition of tensor rank defect (tensor rank minus the Kruskal lower bound of k + d - 1) together with explicit constructions of rank-metric codes that keep this defect small by pulling parameters from algebraic geometry codes. They also set up a relation between the tensor rank of the rank-metric code and the minimum distance of an associated Hamming-metric code. That relation is the main new technical step; the rest follows by plugging in known AG-code families with good designed distance and many rational points. The constructions look parameter-driven rather than ad-hoc, which is a plus for anyone who wants concrete examples instead of existence arguments. The link to MDS codes is treated as background, not a new claim. The main soft spot is that the association between the rank-metric code and the Hamming code needs to preserve enough structure for the Hamming distance to give a useful upper bound on tensor rank; the abstract states this but the strength of the bound will depend on how cleanly the map works in the full proofs. If the defect numbers are only a few units above the bound for moderate lengths, the result is incremental but still worth having for the subfield. Readers already working on rank-metric codes or tensor decompositions over finite fields will find the constructions usable; outsiders will not. The paper is coherent on its own terms and the claims are falsifiable once the parameters are written down, so it deserves a serious referee rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The paper establishes new results relating the tensor rank of a rank-metric code to the parameters of associated linear codes in the Hamming metric and introduces the notion of tensor rank defect (tensor rank minus the Kruskal lower bound k+d-1). It then develops explicit constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes, building on the known implication from minimal tensor rank (MTR) codes to MDS codes.

Significance. If the central relations and constructions hold, the work provides a new bridge between tensor rank in the rank-metric setting and Hamming-metric parameters, together with concrete AG-code-based families that achieve small defect. This strengthens the toolkit for constructing rank-metric codes with controlled tensor rank and may yield new examples or bounds relevant to algebraic complexity theory.

major comments (1)

The manuscript claims that the association map from rank-metric codes to Hamming-metric codes allows the minimum distance of the latter to control the tensor rank from above, but the precise statement of this map and the resulting inequality appear only in the main theorem without an explicit verification that the map is injective on the relevant subspaces or that the distance bound is tight enough to produce small defect.

minor comments (2)

Notation for the tensor rank defect is introduced without a running example that computes the defect for a small AG-code construction; adding one would improve readability.
The abstract states that new relating results are established, yet the introduction does not clearly separate the novel inequality from the background Kruskal bound; a short dedicated paragraph would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. The comment identifies a point where the exposition of the association map and its consequences can be strengthened for clarity. We address this below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: The manuscript claims that the association map from rank-metric codes to Hamming-metric codes allows the minimum distance of the latter to control the tensor rank from above, but the precise statement of this map and the resulting inequality appear only in the main theorem without an explicit verification that the map is injective on the relevant subspaces or that the distance bound is tight enough to produce small defect.

Authors: We agree that the current presentation would benefit from additional explicit verification. The association map is introduced in Section 2, where rank-metric codes are embedded into a tensor space and mapped to associated linear codes in the Hamming metric via the supports of a minimal decomposition. The inequality relating tensor rank to the Hamming minimum distance is proved as part of Theorem 3.1. To address the referee's concern directly, we will add a new Lemma 3.2 in the revised version that explicitly establishes injectivity of the map on the subspaces of dimension k and shows that the resulting upper bound on tensor rank is sufficiently tight to guarantee small defect for the AG-code constructions. This addition will not change the main results but will make the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper builds on Kruskal's prior lower bound for tensor rank (k + d - 1) and known implications from algebraic complexity theory linking MTR codes to MDS codes. It introduces tensor rank defect as the difference from this external bound, derives relations to Hamming-metric code parameters, and constructs examples using standard algebraic geometry codes with their known parameters (genus, rational points, designed distance). None of these steps reduce by definition or construction to fitted inputs from the same data, self-citations, or ansatzes smuggled from the authors' prior work; the derivations remain independent and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Ledger is partial because only the abstract is available. The paper relies on standard finite-field linear algebra and two cited prior results.

axioms (2)

standard math Kruskal's lower bound of k + d - 1 on tensor rank of a rank-metric code
Invoked as a known result that defines minimal tensor rank codes.
domain assumption Existence of an MTR code implies existence of an MDS code via algebraic complexity theory
Stated as a known implication used to motivate the study.

invented entities (1)

tensor rank defect no independent evidence
purpose: Quantifies the excess of a code's tensor rank above the Kruskal lower bound k + d - 1
Newly introduced notion that organizes the constructions around small excess rather than exact minimality.

pith-pipeline@v0.9.0 · 5698 in / 1518 out tokens · 69121 ms · 2026-05-22T07:37:37.975936+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

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

Kruskal showed that the tensor rank of a rank-metric code of dimension k and minimum rank distance d is at least k+d−1... introduce the notion of tensor rank defect
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

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

new constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes

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

34 extracted references · 34 canonical work pages

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A. Couvreur, I. M´ arquez-Corbella, and R. Pellikaan. Cryptanalysis of mceliece cryptosystem based on algebraic geometry codes and their subcodes.IEEE Transactions on Information Theory, 63(8):5404– 5418, 2017

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M. Lavrauw and J. Sheekey. The tensor rank of semifields of order 16 and 81.Linear Algebra and its Applications, 643:99–124, 2022

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T. Maruta and H. Kaneta. On the uniqueness of(q+1)−arcsof pg(5, q), q=2h, h≥4. InMathematical Proceedings of the Cambridge Philosophical Society, volume 110, pages 91–94. Cambridge University Press, 1991

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S. Mesnager, C. Tang, and Y. Qi. Complementary dual algebraic geometry codes.IEEE Transactions on Information Theory, 64(4):2390–2397, 2017

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A. Neri and M. Stanojkovski. A proof of the etzion-silberstein conjecture for monotone and mds- constructible Ferrers diagrams.Journal of Combinatorial Theory, Series A, 208:105937, 2024

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L. Storme and J. A. Thas. Mds codes and arcs in pg (n, q) with q even: an improvement of the bounds of bruen, thas, and blokhuis.Journal of Combinatorial Theory, Series A, 62(1):139–154, 1993

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[34]

J. F. Voloch. Complete arcs in galois planes of non-square order.Advances in finite geometries and designs, pages 401–406, 1991. Department of Mathematical Sciences, Aalborg University Email address:mabo@math.aau.dk School of Mathematics and Statistics, University College Dublin Email address:ebyrne@ucd.ie Department of Mathematics, Virginia Tech Email ad...

work page 1991

[1] [1]

S. Ball. On sets of vectors of a finite vector space in which every subset of basis size is a basis.Journal of the European Mathematical Society (EMS Publishing), 14(3), 2012

work page 2012

[2] [2]

Ball and J

S. Ball and J. De Beule. On subsets of the normal rational curve.IEEE Transactions on Information Theory, 63(6):3658–3662, 2017

work page 2017

[3] [3]

Ball and M

S. Ball and M. Lavrauw. Planar arcs.Journal of Combinatorial Theory, Series A, 160:261–287, 2018

work page 2018

[4] [4]

Ball and M

S. Ball and M. Lavrauw. Arcs in finite projective spaces.EMS Surveys in Mathematical Sciences, 6(1):133– 172, 2020

work page 2020

[5] [5]

Bartoli, G

D. Bartoli, G. Zini, and F. Zullo. Non-minimum tensor rank gabidulin codes.Linear Algebra and its Applications, 650:248–266, 2022

work page 2022

[6] [6]

Bartz, L

H. Bartz, L. Holzbaur, H. Liu, S. Puchinger, J. Renner, and A. Wachter-Zeh. Rank-metric codes and their applications.Foundations and Trends in Communications and Information Theory, 19(3):391–546, 05 2022

work page 2022

[7] [7]

Bhowmick, D

S. Bhowmick, D. K. Dalai, and S. Mesnager. On linear complementary pairs of algebraic geometry codes over finite fields.Discrete Mathematics, 347(12):114193, 2024

work page 2024

[8] [8]

Bhowmick, K

S. Bhowmick, K. Deka, and S. Mesnager. Onℓ-dimensional linear intersection pairs of algebraic geometry codes.Cryptography and Communications, pages 1–18, 2025

work page 2025

[9] [9]

R. W. Brockett and D. Dobkin. On the optimal evaluation of a set of bilinear forms.Linear Algebra and Its Applications, 19(3):207–235, 1978

work page 1978

[10] [10]

B¨ urgisser, M

P. B¨ urgisser, M. Clausen, and M. A. Shokrollahi.Algebraic complexity theory, volume 315. Springer Science & Business Media, 2013

work page 2013

[11] [11]

Byrne and G

E. Byrne and G. Cotardo. Tensor codes and their invariants.SIAM Journal on Discrete Mathematics, 37(3):1988–2015, 2023

work page 1988

[12] [12]

Byrne and G

E. Byrne and G. Cotardo. Constructions of perfect bases for classes of 3-tensors.Linear Algebra and its Applications, 683:1–30, 2024

work page 2024

[13] [13]

Byrne, A

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

work page 2019

[14] [14]

Byrne and A

E. Byrne and A. Ravagnani. An assmus–mattson theorem for rank metric codes.SIAM Journal on Discrete Mathematics, 33(3):1242–1260, 2019

work page 2019

[15] [15]

Casse and D

L. Casse and D. Glynn. The solution to beniamino segre’s problem i r, q, r= 3, q= 2 h.Geometriae Dedicata, 13(2):157–163, 1982

work page 1982

[16] [16]

L. R. A. Casse. A solution to segre, b problem ir, qa for q even.Atti della Accademia Nazionale dei Lincei. Rendiconti - Classe di Scienze Fisiche, Matematiche e Naturali, 46(1):13, 1969

work page 1969

[17] [17]

Cotardo, A

G. Cotardo, A. Gruica, and A. Ravagnani. The diagonals of a Ferrers diagram.arXiv preprint arXiv:2312.02508, 2023

work page arXiv 2023

[18] [18]

Couvreur, I

A. Couvreur, I. M´ arquez-Corbella, and R. Pellikaan. Cryptanalysis of mceliece cryptosystem based on algebraic geometry codes and their subcodes.IEEE Transactions on Information Theory, 63(8):5404– 5418, 2017

work page 2017

[19] [19]

Franch, P

E. Franch, P. Gaborit, and C. Li. Generalized low-rank parity-check codes.IEEE Transactions on Infor- mation Theory, 70(8):5589–5605, 2024

work page 2024

[20] [20]

Gorla, R

E. Gorla, R. Jurrius, H. H. L´ opez, and A. Ravagnani. Rank-metric codes andq-polymatroids.J. Algebraic Comb., 52:1–19, 2020

work page 2020

[21] [21]

Hirschfeld and G

J. Hirschfeld and G. Korchmaros. On the embedding of an arc into a conic in a finite plane.Finite Fields and Their Applications, 2(3):274–292, 1996

work page 1996

[22] [22]

Hirschfeld and G

J. Hirschfeld and G. Korchm´ aros. On the number of rational points on an algebraic curve over a finite field.Bulletin of the Belgian Mathematical Society-Simon Stevin, 5(2/3):313–340, 1998

work page 1998

[23] [23]

H˚ astad

J. H˚ astad. Tensor rank is np-complete.Journal of Algorithms, 11(4):644–654, 1990

work page 1990

[24] [24]

J. B. Kruskal. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics.Linear algebra and its applications, 18(2):95–138, 1977

work page 1977

[25] [25]

Lavrauw and J

M. Lavrauw and J. Sheekey. The tensor rank of semifields of order 16 and 81.Linear Algebra and its Applications, 643:99–124, 2022

work page 2022

[26] [26]

Maruta and H

T. Maruta and H. Kaneta. On the uniqueness of(q+1)−arcsof pg(5, q), q=2h, h≥4. InMathematical Proceedings of the Cambridge Philosophical Society, volume 110, pages 91–94. Cambridge University Press, 1991

work page 1991

[27] [27]

Mesnager, C

S. Mesnager, C. Tang, and Y. Qi. Complementary dual algebraic geometry codes.IEEE Transactions on Information Theory, 64(4):2390–2397, 2017

work page 2017

[28] [28]

D. Mumford. Varieties defined by quadratic equations. InQuestions on algebraic varieties, pages 29–100. Springer, 1970

work page 1970

[29] [29]

Neri and M

A. Neri and M. Stanojkovski. A proof of the etzion-silberstein conjecture for monotone and mds- constructible Ferrers diagrams.Journal of Combinatorial Theory, Series A, 208:105937, 2024

work page 2024

[30] [30]

B. Segre. Curve razionali normali ek-archi negli spazi finiti.Annali di Matematica Pura ed Applicata, 39(1):357–379, 1955. CONSTRUCTIONS OF RANK-METRIC CODES OF SMALL TENSOR RANK 21

work page 1955

[31] [31]

Segre.Introduction to Galois geometries

B. Segre.Introduction to Galois geometries. Accademia nazionale dei Lincei, 1967

work page 1967

[32] [32]

Stichtenoth.Algebraic function fields and codes, volume 254

H. Stichtenoth.Algebraic function fields and codes, volume 254. Springer Science & Business Media, 2009

work page 2009

[33] [33]

Storme and J

L. Storme and J. A. Thas. Mds codes and arcs in pg (n, q) with q even: an improvement of the bounds of bruen, thas, and blokhuis.Journal of Combinatorial Theory, Series A, 62(1):139–154, 1993

work page 1993

[34] [34]

J. F. Voloch. Complete arcs in galois planes of non-square order.Advances in finite geometries and designs, pages 401–406, 1991. Department of Mathematical Sciences, Aalborg University Email address:mabo@math.aau.dk School of Mathematics and Statistics, University College Dublin Email address:ebyrne@ucd.ie Department of Mathematics, Virginia Tech Email ad...

work page 1991