arxiv: 2605.12295 · v1 · submitted 2026-05-12 · 🧮 math.CO · cs.IT· math.IT

Recognition: 2 theorem links

· Lean Theorem

Symmetric Tensor Decompositions over Finite Fields

Ferdinando Zullo, Giuseppe Cotardo

Pith reviewed 2026-05-13 03:32 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT

keywords symmetric tensor rankfinite fieldslinearized polynomialsGabidulin codesrank-metric codesbilinear complexityFrobenius automorphismsymmetric decompositions

0 comments

The pith

The symmetric tensor rank of finite field multiplication equals the symmetric tensor rank of its associated one-dimensional Gabidulin code.

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

The paper establishes methods to compute symmetric tensor decompositions for the multiplication map in finite field extensions. It maps symmetric linearized polynomials to symmetric bilinear forms through the field trace, converting the decomposition task into a spanning problem by rank-one polynomials. These spanning conditions become explicit linear systems over finite fields, solved using the Frobenius automorphism for practical criteria. The approach recovers known symmetric bilinear complexities for small extensions and yields explicit decompositions for various parameters. It further defines the symmetric tensor-rank for symmetric rank-metric codes and proves equality with the multiplication map's rank in the case of the natural one-dimensional Gabidulin code linked to field multiplication.

Core claim

Via the field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and matrices. This allows symmetric tensor decompositions of the multiplication map to be reformulated as spanning problems by rank-one symmetric linearized polynomials. The spanning conditions translate into linear systems over finite fields, with effective criteria obtained via the Frobenius automorphism. For the natural one-dimensional Gabidulin code associated with finite field multiplication, the symmetric tensor-rank of the code coincides with the symmetric tensor rank of the multiplication map.

What carries the argument

The identification of symmetric linearized polynomials with symmetric bilinear forms via the field trace, which recasts tensor decompositions as spanning problems by rank-one elements and enables translation to linear systems solved with the Frobenius automorphism.

If this is right

Known values of the symmetric bilinear complexity for small extension degrees are recovered through the linear system criteria.
Explicit symmetric decompositions are obtained for several extension parameters.
The symmetric tensor-rank invariant applies to symmetric rank-metric codes and equals the multiplication rank for the associated one-dimensional Gabidulin code.
Computationally effective criteria for the spanning conditions follow from the Frobenius automorphism.

Where Pith is reading between the lines

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

The equivalence could enable bounding tensor ranks using properties of rank-metric codes in broader algebraic settings.
The linear system approach may extend to decompositions of other bilinear maps over finite fields.
Explicit decompositions obtained this way could support constructions in applications relying on efficient finite field arithmetic.

Load-bearing premise

The identification of symmetric linearized polynomials with symmetric bilinear forms via the field trace preserves the relevant spanning and rank properties without introducing extra dependencies or losing information.

What would settle it

A specific small extension degree where the minimal number of rank-one symmetric linearized polynomials needed to span the space differs from the length of the shortest symmetric decomposition of the multiplication map obtained by direct search.

read the original abstract

We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing symmetric tensor decompositions to be reformulated as spanning problems by rank-one symmetric linearized polynomials. We translate these spanning conditions into explicit linear systems over finite fields and use the Frobenius automorphism to obtain computationally effective criteria. As applications, we recover known values of the symmetric bilinear complexity for small extension degrees and obtain explicit symmetric decompositions for several parameters. We also introduce the symmetric tensor-rank of a symmetric rank-metric code and show that, for the natural one-dimensional Gabidulin code associated with finite field multiplication, this invariant coincides with the symmetric tensor rank of the multiplication map.

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 turns symmetric tensor rank for field multiplication into linear systems over linearized polynomials, with a code invariant that matches for 1D Gabidulin but follows directly from the definition.

read the letter

The main takeaway is that this work recasts symmetric tensor decompositions of finite field multiplication as spanning problems by rank-one symmetric linearized polynomials, then converts those into explicit linear systems over the base field with Frobenius criteria for solving them. They recover known symmetric bilinear complexity numbers for small extension degrees and produce concrete decompositions for several parameters. They also define a symmetric tensor rank for symmetric rank-metric codes and show it equals the multiplication map rank for the natural one-dimensional Gabidulin code generated by that map. The trace identification between symmetric linearized polynomials and symmetric bilinear forms rests on standard finite field properties and appears to preserve the necessary rank and spanning relations without obvious loss or extra dependencies. The linear-system translation and Frobenius criteria are the practical pieces that could help with computation beyond what was easy before. The applications line up with prior results, which is reassuring rather than surprising. The Gabidulin coincidence is correct but holds almost by construction: once the code rank is defined via minimal spanning sets of rank-one elements, any such set for a one-dimensional code directly yields a decomposition of the generator up to scalar. That part does not require the full machinery of the linear systems or Frobenius criteria to see. No load-bearing circularity or shaky assumptions show up in the abstract or the described claims. The work stays grounded in existing tools from finite fields and rank-metric codes. This is for people who need concrete methods to bound or compute symmetric bilinear complexity over finite fields, or who work with rank-metric codes and want a new invariant to play with. A reader focused on fast arithmetic or coding constructions would get usable criteria and examples. It is worth sending to referees because the reformulation is technically clear, the small-case checks are consistent, and the tools are reproducible in principle.

Referee Report

2 major / 2 minor

Summary. The manuscript studies symmetric tensor decompositions of multiplication maps over finite field extensions via linearized polynomials. Symmetric linearized polynomials are identified with symmetric bilinear forms and matrices through the field trace, reformulating decompositions as spanning problems by rank-one elements. These spanning conditions are translated into explicit linear systems over finite fields, with Frobenius automorphism yielding effective criteria. Applications recover known symmetric bilinear complexity values for small extension degrees, provide explicit decompositions for several parameters, and introduce the symmetric tensor-rank of a symmetric rank-metric code, showing coincidence with the multiplication map's rank for the natural one-dimensional Gabidulin code.

Significance. If the identifications and translations hold, the linear-system and Frobenius criteria offer a practical computational framework for symmetric tensor ranks over finite fields, with the recovery of known small-degree values providing consistency checks and the explicit decompositions adding concrete value. The new invariant for symmetric rank-metric codes is a conceptual contribution, though its demonstrated utility is limited to the 1D Gabidulin case.

major comments (2)

[The application introducing the symmetric tensor-rank of symmetric rank-metric codes and the Gabidulin coincidence (near] The claimed coincidence that the symmetric tensor-rank of the natural one-dimensional Gabidulin code equals the symmetric tensor rank of the multiplication map follows immediately from the definition of the code invariant (minimal spanning set of rank-1 symmetric linearized polynomials) once the code is recognized as 1-dimensional; any minimal spanning set for the code subspace directly yields a decomposition of the generator up to scalar. This observation does not rely on the trace identification, linear-system translation, or Frobenius criteria. The manuscript should clarify what non-trivial content the stated theorem adds beyond the definition.
[Section on the trace identification and reformulation as spanning problems] The central reformulation rests on the identification of symmetric linearized polynomials with symmetric bilinear forms via the field trace preserving spanning sets and rank-1 properties without extra dependencies or information loss. An explicit proof or verification that this map is bijective on the relevant spaces and rank-preserving would be required to support all subsequent linear-system translations.

minor comments (2)

A table summarizing the recovered known values (with references) and the parameters for which explicit decompositions are constructed would improve readability and allow quick assessment of the applications.
Early and consistent definition of notation for linearized polynomials, the trace map, and the new code invariant would aid readers unfamiliar with the combination of tensor rank and rank-metric coding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and will incorporate revisions to improve the clarity and rigor of the manuscript.

read point-by-point responses

Referee: [The application introducing the symmetric tensor-rank of symmetric rank-metric codes and the Gabidulin coincidence (near] The claimed coincidence that the symmetric tensor-rank of the natural one-dimensional Gabidulin code equals the symmetric tensor rank of the multiplication map follows immediately from the definition of the code invariant (minimal spanning set of rank-1 symmetric linearized polynomials) once the code is recognized as 1-dimensional; any minimal spanning set for the code subspace directly yields a decomposition of the generator up to scalar. This observation does not rely on the trace identification, linear-system translation, or Frobenius criteria. The manuscript should clarify what non-trivial content the stated theorem adds beyond the definition.

Authors: We agree that the equality is a direct consequence of the definitions once the code is recognized as one-dimensional. The non-trivial contribution of this part of the work lies in introducing the symmetric tensor-rank invariant for symmetric rank-metric codes in the first place and in identifying the natural one-dimensional Gabidulin code as the appropriate object to which this invariant applies in the context of field multiplication. We will revise the manuscript to explicitly note the definitional character of the equality and to better highlight the conceptual novelty of the new invariant. revision: yes
Referee: [Section on the trace identification and reformulation as spanning problems] The central reformulation rests on the identification of symmetric linearized polynomials with symmetric bilinear forms via the field trace preserving spanning sets and rank-1 properties without extra dependencies or information loss. An explicit proof or verification that this map is bijective on the relevant spaces and rank-preserving would be required to support all subsequent linear-system translations.

Authors: We acknowledge that an explicit verification strengthens the exposition. Although the trace-based identification between symmetric linearized polynomials and symmetric bilinear forms is standard, we will add a dedicated lemma in the revised manuscript that proves the map is bijective on the relevant vector spaces and that it preserves rank-one elements together with the property of spanning sets, thereby justifying the subsequent translations to linear systems. revision: yes

Circularity Check

1 steps flagged

The claimed coincidence for the 1D Gabidulin code reduces to the definition of code tensor-rank via spanning by rank-1 elements.

specific steps

self definitional [Abstract (final sentence) and the section introducing the code invariant]
"We also introduce the symmetric tensor-rank of a symmetric rank-metric code and show that, for the natural one-dimensional Gabidulin code associated with finite field multiplication, this invariant coincides with the symmetric tensor rank of the multiplication map."

The definition of the code's symmetric tensor-rank is the minimal number of rank-1 symmetric linearized polynomials needed to span the code subspace. For a 1-dimensional code generated by the multiplication map, this number is exactly the symmetric tensor rank of the map itself (up to scalar). The equality therefore holds by construction of the definition and the choice of code, without requiring the trace map, linear systems, or Frobenius criteria developed elsewhere in the paper.

full rationale

The paper's central application result is that the newly introduced symmetric tensor-rank of a symmetric rank-metric code coincides with the tensor rank of the multiplication map for the natural 1D Gabidulin code. Because the code is 1-dimensional and spanned by the map itself, any minimal spanning set of rank-1 symmetric linearized polynomials for the code subspace is definitionally a decomposition of the generator (up to scalar). The trace identification, linear-system translation, and Frobenius criteria are not needed for this observation, making the coincidence tautological once the invariant is defined via the spanning reformulation. The remainder of the paper (reformulations and explicit decompositions for small degrees) does not rely on this step and appears self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard finite-field axioms (existence of trace, Frobenius automorphism, vector-space structure) and introduces one new entity: the symmetric tensor rank of a symmetric rank-metric code. No free parameters are fitted; all criteria are derived from the linear-algebra reformulation.

axioms (2)

standard math The trace map from a finite field extension to its base field is a non-degenerate symmetric bilinear form.
Invoked to identify symmetric linearized polynomials with symmetric bilinear forms and matrices.
standard math The Frobenius automorphism generates the Galois group and acts linearly on the vector space.
Used to obtain computationally effective criteria from the linear systems.

invented entities (1)

symmetric tensor-rank of a symmetric rank-metric code no independent evidence
purpose: New invariant measuring the minimal number of rank-one symmetric terms needed to span the code
Defined in the paper; shown to coincide with the multiplication rank for the one-dimensional Gabidulin code.

pith-pipeline@v0.9.0 · 5416 in / 1514 out tokens · 80324 ms · 2026-05-13T03:32:10.977341+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/RealityFromDistinction.lean reality_from_one_distinction unclear
We also introduce the symmetric tensor-rank of a symmetric rank-metric code and show that, for the natural one-dimensional Gabidulin code associated with finite field multiplication, this invariant coincides with the symmetric tensor rank of the multiplication map.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing symmetric tensor decompositions to be reformulated as spanning problems by rank-one symmetric linearized polynomials.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Alnajjarine and M

N. Alnajjarine and M. Lavrauw. Linear complete symmetric rank-distance codes.De- signs, Codes and Cryptography, 94(2):36, 2026. 19

work page 2026
[2]

Ballet, J

S. Ballet, J. Pieltant, M. Rambaud, H. Randriambololona, R. Rolland, and J. Chau- mine. On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry.Russian Mathematical Surveys, 76(1):29, 2021

work page 2021
[3]

Bik and A

A. Bik and A. Neri. Higher-degree symmetric rank-metric codes.SIAM Journal on Applied Algebra and Geometry, 8(4):931–967, 2024

work page 2024
[4]

B¨ urgisser, M

P. B¨ urgisser, M. Clausen, and M. A. Shokrollahi.Algebraic complexity theory, volume

work page
[5]

Springer Science & Business Media, 2013

work page 2013
[6]

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
[7]

Byrne, G

E. Byrne, G. Longobardi, and R. Trombetti. q-polymatroids associated with restricted rank-metric codes.arXiv preprint arXiv:2602.04080, 2026

work page arXiv 2026
[8]

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
[9]

Chaumine

J. Chaumine. On the bilinear complexity of the multiplication in small finite fields. Comptes Rendus de l’Acad´ emie des Sciences, S´ erie I, 343(4):265–266, 2006

work page 2006
[10]

D. V. Chudnovsky and G. V. Chudnovsky. Algebraic complexities and algebraic curves over finite fields.Journal of Complexity, 4(4):285–316, 1988

work page 1988
[11]

H. F. de Groote. Characterization of division algebras of minimal rank and the structure of their algorithm varieties.SIAM Journal on Computing, 12(1):101–117, 1983

work page 1983
[12]

Delsarte

P. Delsarte. Bilinear forms over a finite field, with applications to coding theory.Journal of Combinatorial Theory, Series A, 25(3):226–241, 1978

work page 1978
[13]

Gabidulin and N

E. Gabidulin and N. Pilipchuk. Representation of a finite field by symmetric matrices and applications. InProc: 8th Int. Workshop on Algebraic and Combinatorial Coding Theory. pp. 120–123. Tsarskoe Selo, Russia (2002)

work page 2002
[14]

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

work page 1985
[15]

E. M. Gabidulin and N. I. Pilipchuk. Symmetric rank codes.Problems of Information Transmission, 40(2):103–117, 2004

work page 2004
[16]

E. M. Gabidulin and N. I. Pilipchuk. Symmetric matrices and codes correcting rank errors beyond the⌊(d−1)/2⌋bound.Discrete applied mathematics, 154(2):305–312, 2006

work page 2006
[17]

W. K. Kadir, C. Li, and F. Zullo. Encoding and decoding of several optimal rank metric codes.Cryptography and Communications, 14(6):1281–1300, 2022

work page 2022
[18]

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
[19]

Lidl and H

R. Lidl and H. Niederreiter.Finite fields. Number 20. Cambridge university press, 1997

work page 1997
[20]

Longobardi, G

G. Longobardi, G. Lunardon, R. Trombetti, and Y. Zhou. Automorphism groups and new constructions of maximum additive rank metric codes with restrictions.Discrete Mathematics, 343(7):111871, 2020

work page 2020
[21]

Mushrraf and F

U. Mushrraf and F. Zullo. On perfect symmetric rank-metric codes.Archiv der Math- ematik, 125(3):259–271, 2025

work page 2025
[22]

O. Ore. On a special class of polynomials.Transactions of the American Mathematical Society, 35(3):559–584, 1933

work page 1933
[23]

K.-U. Schmidt. Symmetric bilinear forms over finite fields of even characteristic.Journal of Combinatorial Theory, Series A, 117(8):1011–1026, 2010

work page 2010
[24]

K.-U. Schmidt. Symmetric bilinear forms over finite fields with applications to coding theory.Journal of Algebraic Combinatorics, 42(2):635–670, 2015

work page 2015
[25]

Seroussi and A

G. Seroussi and A. Lempel. On symmetric algorithms for bilinear forms over finite fields.Journal of Algorithms, 5(3):327–344, 1984. 20

work page 1984
[26]

M. A. Shokrollahi. Optimal algorithms for multiplication in certain finite fields using elliptic curves.SIAM Journal on Computing, 21(6):1193–1198, 1992

work page 1992
[27]

Tang and Y

W. Tang and Y. Zhou. A new family of maximum linear symmetric rank-distance codes.Finite Fields and Their Applications, 111:102787, 2026

work page 2026
[28]

Wan.Geometry of matrices: in memory of professor LK Hua (1910–1985)

Z.-X. Wan.Geometry of matrices: in memory of professor LK Hua (1910–1985). World Scientific, 1996

work page 1910
[29]

Wu and Z

B. Wu and Z. Liu. Linearized polynomials over finite fields revisited.Finite Fields and Their Applications, 22:79–100, 2013

work page 2013
[30]

Y. Zhou. On equivalence of maximum additive symmetric rank-distance codes.Designs, Codes and Cryptography, 88(5):841–850, 2020. 21

work page 2020