Partially scattered linearized polynomials and rank metric codes

Corrado Zanella; Giovanni Longobardi

REVIEW 2 major objections 4 minor 25 references

Partially scattered linearized polynomials and rank metric codes

T0 review · 2 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read The paper proves that a polynomial being R-$q^t$-partially scattered is exactly the condition under which a whole family of rectangular rank-metric codes attains the Singleton bound.

desk verdict A sound and useful unification of scattered polynomial notions; the central iff with MRD codes holds, with a few fixable presentational gaps. read the letter →

arxiv 2009.11537 v3 pith:RQYAOS25 submitted 2020-09-24 math.CO

Giovanni Longobardi , Corrado Zanella This is my paper · ORCID

classification math.CO MSC 51E2005B2551E22

keywords linearizedpolynomialpartiallyscatteredlinearsetfiniteprojectivespacerankmetriccodeMRD-codesubgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces two weakenings of the classical notion of a scattered linearized polynomial over a finite field, calling them L-$q^t$-partially scattered and R-$q^t$-partially scattered, and proves that the right-handed version is precisely the certification that a natural family of rectangular rank-metric codes is optimal. A linearized polynomial $f(x)$ acts as an $\mathbb F_q$-linear map on $\mathbb F_{q^n}$; for a proper divisor $t$ of $n$, the R-condition says that if $f(y)/y=f(z)/z$ and $y/z\in\mathbb F_{q^t}$, then in fact $y/z\in\mathbb F_q$. The central theorem shows that this condition holds exactly when every code $C_{f,\sigma,k}=\langle f_{t,\sigma}(x),x^{q^{kt}}\rangle_{\mathbb F_{q^n}}$ is an MRD-code with parameters $(n,t,q;t-1)$, where $f_{t,\sigma}$ is the restriction of $f(\sigma x)$ to $\mathbb F_{q^t}$. This matters because one algebraic condition on a single polynomial produces many provably optimal codes at once, and the classical scattered polynomials reappear as the intersection of the left and right conditions.

What carries the argument

The engine of the paper is the difference map $f_\rho(x)=f(\rho x)-\rho f(x)$. Proposition 2.6 asserts that $f$ is R-$q^t$-partially scattered exactly when $f_\rho$ is bijective for every $\rho\in\mathbb F_{q^t}\setminus\mathbb F_q$, and L-$q^t$-partially scattered exactly when $f_\rho$ is bijective for every $\rho\in\mathbb F_{q^n}\setminus\mathbb F_{q^t}$. This converts the ratio definition into a bijectivity statement that is preserved under adding $q^t$-polynomials, and it is this preservation that drives the MRD proof. The companion construction is the code $C_{f,\sigma,k}$: its generators are the $n\times t$ matrix of the $\mathbb F_q$-linear restriction $f_{t,\sigma}(x)=f(\sigma x)$ on $\mathbb F_{q^t}$ and the matrix of $x^{q^{kt}}$. The MRD bound is reached exactly when every nonzero element of this span has rank at least $t-1$, which is what R-partial scatteredness supplies.

What would settle it

A concrete finite-field check can settle the theorem's converse direction. For $q=3$, $n=6$, $t=3$, take the polynomial $f(x)=T_1(x)+U_5(x)$ from the paper's construction, where $T_1(x)=(\operatorname{Tr}_{\mathbb F_{3^6}/\mathbb F_{3^3}}(x))^3$ and $U_5(x)=(x-x^{3^3})^{3^5}$; Proposition 2.18 says this is R-$q^3$-partially scattered. Compute the minimum rank among all nonzero $3^{12}$ elements of the code $\langle f_{t,1}(x),x\rangle_{\mathbb F_{3^6}}$, viewed as $6\times3$ matrices over $\mathbb F_3$. The theorem predicts every nonzero element has rank at least $2$; a single rank-$1$ element would refute it.

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Extended reading notes

Core claim

On the paper's own terms, the finding is Theorem 3.3(a). Let $f(x)\in\mathbb F_{q^n}[x]$ be a $q$-polynomial and let $t$ be a nontrivial divisor of $n$. Then $f$ is R-$q^t$-partially scattered if and only if, for every $\sigma\in\mathbb F_{q^n}^*$ and every integer $k\ge0$, the code $C_{f,\sigma,k}$ spanned over $\mathbb F_{q^n}$ by the restriction $x\mapsto f(\sigma x)$ on $\mathbb F_{q^t}$ together with the monomial $x^{q^{kt}}$ is an MRD-code with parameters $(n,t,q;t-1)$. The forward direction works because adding any scalar multiple of $x^{q^{kt}}$ to $f(\sigma x)$ preserves the R-partially-scattered property, so every nonzero element of the code restricts to $\mathbb F_{q^t}$ with kernel of dimension at most one; the code therefore has the right dimension $2n$ and minimum distance at least $t-1$. The converse recovers the ratio condition from the same rank constraint. In the square case $t=t'=2$, the same equivalence is shown for the single code $\langle f(x),x,x^{q^2}\rangle_{\mathbb F_{q^4}}$ with parameters $(4,4,q;2)$.

Load-bearing premise

The load-bearing premise is Proposition 2.6, stated without proof: the partial-scattered ratio conditions can be replaced by the bijectivity of $f(\rho x)-\rho f(x)$ for every $\rho$ outside the relevant subfield, and if that equivalence failed, the whole MRD characterization in Theorem 3.3 would collapse.

Editorial extensions

If this is right

Every R-$q^t$-partially scattered polynomial gives, for each $\sigma\in\mathbb F_{q^n}^*$ and each $k\ge0$, an MRD-code $C_{f,\sigma,k}$ with parameters $(n,t,q;t-1)$, so a single polynomial certifies many optimal codes.
Classical scattered polynomials are exactly the intersection of the L- and R-conditions, so the known theory of scattered linearized polynomials is included as the special case; scattered polynomials now also yield rectangular MRD-codes, not only square ones.
When $\gcd(t,t')=1$, an L-$q^t$-partially scattered polynomial is automatically R-$q^{t'}$-partially scattered, so the L-condition alone produces MRD-codes after swapping the roles of $t$ and $t'$.
The paper's binomial and trinomial examples, with explicit norm or trace conditions, provide new R-partially scattered polynomials and hence explicit new families of rectangular MRD-codes.
For $t=t'=2$, the R-condition is equivalent to the single code $\langle f(x),x,x^{q^2}\rangle_{\mathbb F_{q^4}}$ being MRD with distance 2, and such codes have both idealisers equal to $\mathbb F_{q^4}$ and are equivalent to generalized Gabidulin codes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because Proposition 2.6 reduces partial scatteredness to bijectivity of $f_\rho$ for finitely many $\rho$, the paper implicitly offers a computational certificate for a polynomial to generate MRD-codes: test the determinant of the associated matrix of $f_\rho$ for each $\rho$ in the relevant set instead of checking ratio conditions on all pairs $(y,z)$.
The construction suggests a hierarchy across divisors: the same $f$ can be R-$q^t$-partially scattered for several $t$'s, and Proposition 2.7 shows that L-scatteredness at one divisor can become R-scatteredness at a coprime one, so a single polynomial may certify MRD-codes of several different rectangular shapes.
The square case $t=t'=2$ identifies R-$q^2$-partial scatteredness with a single 4x4 MRD-code with maximum idealisers, making the condition a concrete target for classifying MRD-codes that are not generalized Gabidulin.
An open question implicit in the paper is whether the kernel-dimension bound $t'$ in Proposition 3.1(c) is sharp; finding an R-partially scattered polynomial with a larger kernel would give an MRD-code not explained by the composition construction, and the rectangular setting may be the right place to look.

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Formalized claims in Lean

Claim #1: On the paper's own terms, the finding is Theorem 3.3(a). Let $f(x)\in\mathbb F_{q^n}[x]$ be a $q$-polynomial and let $t$ be a nontrivial divisor of $n$. Then $f$ is R-$q^t$-partially scattered if and only if, for every $\sigma\in\mathbb F_{q^n}^*$ and every integer $k\ge0$, the code $C_{f,\sigma,k}$ spanned over $\mathbb F_{q^n}$ by the restriction $x\mapsto f(\sigma x)$ on $\mathbb F_{q^t}$ toget

/-- @claim 1 On the paper's own terms, the finding is Theorem 3.3(a). Let $f(x)\in\mathbb F_{q^n}[x]$ be a $q$-polynomial and let $t$ be a nontrivial divisor of $n$. Then $f$ is R-$q^t$-partially scattered if and only if, for every $\sigma\in\mathbb F_{q^n}^*$ and every integer $k\ge0$, the code $C_{f,\sigma,k}$ spanned over $\mathbb F_{q^n}$ by the restriction $x\mapsto f(\sigma x)$ on $\mathbb F_{q^t}$ toget -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague — read this one if you care about scattered polynomials or MRD codes. The new L/R-partially scattered split is a real contribution: it explains scatteredness as the intersection of two weaker properties, and Theorem 3.3 turns the R-property into a clean iff for a whole family of rectangular MRD codes. I checked the central derivations and they are sound. Proposition 2.6 is the engine and it is true, though the proof is omitted; the dimension check in Theorem 3.3(a) is also skipped but follows quickly. The geometric side (Theorem 2.3, including the t=2 canonical subgeometry statement) is well done and the examples in Section 2.3 give the definitions life. The paper is honest about what is new and correctly distinguishes its rectangular treatment from earlier square-code results.

Soft spots are minor and fixable. Proposition 2.20's printed adjoint step is invalid as written, but the statement is true and a short correct proof exists. The GAP examples in Remarks 2.13 and 2.18 are not accompanied by scripts, so those computational claims are not independently checkable. The kernel criterion used in Proposition 2.16 is asserted without proof or citation. Proposition 2.6, the most load-bearing unproved item, should get a proof in revision; it is short. There is no circularity: the definitions are new, theorems follow from them, and the self-citations contribute examples and background.

Verdict: this deserves peer review, and I would accept it after a minor revision that fills those gaps. The right audience is finite geometry and coding theory; a broader reading group might skim the examples and keep the main equivalence. For myself, I would cite it if I write about scattered polynomials or MRD codes.

Referee Report

2 major / 4 minor

Summary. The paper introduces two weakenings of scattered linearized polynomials over F_{q^n}, for a nontrivial divisor t of n: L-q^t-partially scattered and R-q^t-partially scattered polynomials. It shows that a polynomial is scattered exactly when it is both L- and R-partially scattered. The paper gives geometric characterizations in terms of the linear sets R_f and L_f, algebraic criteria in terms of the bijectivity of the maps f_rho(x)=f(rho x)-rho f(x), several families of examples, and a duality result. The main structural result, Theorem 3.3(a), proves that f is R-q^t-partially scattered if and only if for every sigma and every k the F_{q^n}-span C_{f,sigma,k}=<f_{t,sigma}(x),x^{q^{kt}}> is an MRD-code with parameters (n,t,q;t-1). The paper also studies idealisers of these codes and gives a t=t'=2 specialization.

Significance. The paper's central equivalence is a meaningful contribution: it turns the R-partially-scattered condition into an exact certification that a full family of rectangular rank-metric codes is optimal, and it places classical scattered polynomials inside the intersection of the L- and R-properties. The main arguments are sound; I checked the key steps, including the algebraic criterion in Proposition 2.6, the geometric equivalences in Theorem 2.3, and the MRD direction in Theorem 3.3. The omitted details are repairable and do not affect the truth of the claims. The paper also provides useful constructions and examples, including explicit binomial and trinomial families and GAP-based computational observations, and it correctly identifies the adjoint as a self-dual operation for the partial-scattered properties.

major comments (2)

[§2.2, Proposition 2.6] Proposition 2.6 is stated without proof, yet it is load-bearing: it is used in Proposition 2.9, Proposition 2.10, and in the R-to-MRD direction of Theorem 3.3(a). The statement is true, but the manuscript should contain a short direct proof. For example, for part (a), if f_rho has a nontrivial kernel for some rho in F_{q^t}\F_q, choose nonzero y with f(rho y)=rho f(y); then f(y)/y=f(rho y)/(rho y) with ratio rho outside F_q, contradicting (4); the converse is immediate. Please add a proof or an explicit reference.
[§3, Theorem 3.3(a), Eq. (14)] The proof of Theorem 3.3(a) omits the dimension check that C_{f,sigma,k} has F_q-dimension 2n, which is required for the Singleton-like bound. This gap is readily filled: if f is R-q^t-partially scattered, the displayed F_{q^n}-span has dimension 2n because a relation f(sigma x)=c x^{q^{kt}} on F_{q^t} would imply, for beta in F_{q^t}\F_q, that f(sigma beta)/(sigma beta)=f(sigma)/sigma with ratio beta in F_{q^t}\F_q, contradicting (4). Please include this argument in the proof.

minor comments (4)

[§2.1, Theorem 2.3(iii)] The proof invokes the fact that for every point in PG(2t'-1,q^2) there is at least one (q+1)-secant line to a canonical F_q-subgeometry, but this fact is neither proved nor cited; please add a reference.
[§2.3, Proposition 2.16] The equivalence that g(x)=x+x^{q^t}+delta x^{q^{2t}} is bijective iff Tr_{q^{3t}/q^t}(delta)-N_{q^{3t}/q^t}(delta) != 2 is asserted without proof or citation; since this is a standard determinant criterion for the Dickson matrix, a reference would suffice.
[§2.4, Proposition 2.20] The proof contains an invalid step as written: from hat{f}(lambda y)=lambda hat{f}(y) it does not follow by applying the adjoint map to both sides that lambda f(y)=f(lambda y), because the adjoint is not multiplicative in the displayed sense. The duality statement is nevertheless true, for instance via (f_rho)^* = -(f^*)_rho together with Proposition 2.6. Please replace the proof with a correct argument.
[Throughout] There are several small typographical issues, e.g. 'hyphoteses' before Remark 3.6 and 'N q2t/qt' without braces in Remark 2.13; please proofread carefully.

Circularity Check

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No circularity: the R/L partial-scattered definitions are postulates and Theorem 3.3 is proved from them; self-citations appear only in illustrative examples.

full rationale

The central derivation chain is self-contained. Definitions 2.1 and 2.2 are genuine postulates, and the paper proves Theorem 2.3, Proposition 2.6, Propositions 2.9–2.10, Proposition 3.1, and Theorem 3.3 from these definitions rather than importing the target result. Proposition 2.6 is stated without proof, but it is an elementary algebraic reformulation of (3)–(4), not an imported theorem, so its omission is an exposition gap, not circularity. Theorem 3.3(a) does not define R-partial scatteredness in terms of MRD codes: the forward direction uses Proposition 2.9 and the fact that a rank drop on F_{q^t} would produce elements satisfying f(y)/y = f(z)/z with y/z in F_{q^t}\F_q, contradicting (4); the converse uses the MRD hypothesis to force the kernel of f(u)x + m_u x^{q^{kt}} to have dimension at most one, hence λ in F_q in (16). The F_q-dimension 2n of the code is a direct span fact, not a fitted parameter. The cited results [6] and [14], which include a current author, appear only in remarks and examples and are not load-bearing for the main equivalence; the load-bearing citations [7], [18], [22], and [23] are external or classical. Proposition 2.20 contains an incorrect printed adjoint step, but the duality statement is independently derivable via Proposition 2.6, so it is a presentation flaw rather than a circular reduction. No fitted parameters are renamed as predictions, and no uniqueness theorem by the present authors is invoked to force a conclusion. Thus the paper has no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

free_parameters: none. The paper contains no fitted or hand-chosen numerical parameters; t, t', s, delta are variables in theorems with explicit hypotheses (e.g., N(delta) != 1), not fitted values. axioms: six domain assumptions drawn from the cited literature (Lunardon-Polverino projection, Blokhuis-Lavrauw bound, Delsarte duality, the [7] classification, the Dickson determinant criterion for x + x^{q^t} + delta x^{q^{2t}}, and the known scatteredness of LP/Sheekey-type binomials), plus one standard_math entry for the representation of endomorphisms by linearized polynomials and the Dickson-matrix rank formula. invented_entities: none. The new objects (L- and R-partially scattered polynomials, the codes C_{f,sigma,k}) are definitions and constructions whose properties are proved, not entities postulated to force a conclusion.

assumptions (7)

domain assumption Lunardon-Polverino projection theorem: every F_q-linear set spanning a subspace is the projection of a canonical F_q-subgeometry
Invoked in Section 1 and in the proof of Theorem 2.3(iii) (cited [16], [17]) to represent R_f as a projection and to conclude it is a subgeometry when the vertex is empty.
domain assumption Blokhuis-Lavrauw bound: a subspace scattered with respect to a spread has dimension at most half the ambient dimension when the latter is even
Used in Section 1 and in Proposition 2.4 to identify maximum scattered linear sets (cited [4]).
domain assumption Delsarte dual of an F_q-linear MRD code with minimum distance d > 1 is again MRD
Used in Theorem 3.3(b) and Proposition 3.4 to control C_perp (cited [10]).
domain assumption Classification of MRD codes with both idealisers F_{q^n}: for 2 <= n <= 6 or n = 9 they are generalized Gabidulin codes
Carries Proposition 3.4's claim that the (4,4,q;2) code C is Gabidulin-equivalent (cited [7, Theorem 1.1]).
domain assumption For g(x) = x + x^{q^t} + delta x^{q^{2t}} over F_{q^{3t}}, g is bijective iff Tr_{q^{3t}/q^t}(delta) - N_{q^{3t}/q^t}(delta) != 2
Invoked without proof or citation in Proposition 2.16 to build the F_{q^{3t}} example family; a Dickson-determinant computation.
domain assumption The binomials delta x^{q^s} + x^{q^{t+s}} (LP/Sheekey type) are scattered for suitable norms, e.g. N_{q^{2t}/q^t}(delta) = -1 (t=4) and N_{q^{2t}/q}(delta) not in {0,1}
Used in Remark 2.13 (citing [6, Theorem 7.2]) and Remark 3.6 as input examples; results by overlapping authors but independently published.
standard math Linearized polynomials of q-degree less than n represent End_{F_q}(F_{q^n}) uniquely, and the rank of f equals the rank of its Dickson matrix
Background used throughout, cited to [12], [24]; standard finite-field theory.

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Pith. "Pith review of Partially scattered linearized polynomials and rank metric codes." pith.science (2026). https://pith.science/paper/RQYAOS25

@misc{pith2026200911537,
  author       = {Pith},
  title        = {Pith review of: Partially scattered linearized polynomials and rank metric codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQYAOS25}},
  note         = {Machine review of arXiv:2009.11537}
}

abstract

A linearized polynomial $f(x)\in\mathbb F_{q^n}[x]$ is called scattered if for any $y,z\in\mathbb F_{q^n}$, the condition $zf(y)-yf(z)=0$ implies that $y$ and $z$ are $\mathbb F_{q}$-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are defined and investigated. Let $t$ be a nontrivial positive divisor of $n$. By weakening the property defining a scattered linearized polynomial, L-$q^t$-partially scattered and R-$q^t$-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-$q^t$- and R-$q^t$-partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited.

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Reference graph

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Bartoli, M

D. Bartoli, M. Giulietti, G. Marino, O. Polverino: Maxim um Scattered Linear Sets and Complete Caps in Galois Spaces, Combinatori ca 38 (2018), 255–278

work page 2018

[2] [2]

Bartoli, C

D. Bartoli, C. Zanella, F. Zullo: A new family of maximum s cattered linear sets in PG(1 ,q 6), Ars Math. Contemp. 19 (2020), 125–145

work page 2020

[3] [3]

Bartoli, Y

D. Bartoli, Y. Zhou: Exceptional scattered polynomials , J. Algebra 509 (2018), 507–534,

work page 2018

[4] [4]

Blokhuis, M

A. Blokhuis, M. Lavrauw: Scattered spaces with respect t o a spread in PG(n,q ), Geom. Dedicata 81 (2000), 231–243

work page 2000

[5] [5]

Csajb´ ok, G

B. Csajb´ ok, G. Marino, O. Polverino: Classes and equiva lence of linear sets in PG(1,q n), J. Combin. Theory Ser. A 157 (2018), 402–426

work page 2018

[6] [6]

Csajb´ ok, G

B. Csajb´ ok, G. Marino, O. Polverino, C. Zanella: A new fa mily of MRD-codes, Linear Algebra Appl. 548 (2018), 203–220

work page 2018

[7] [7]

Csajb´ ok, G

B. Csajb´ ok, G. Marino, O. Polverino, Y. Zhou: MRD codes w ith maximum idealizers, Discrete Math. 343 (2020), 111985

work page 2020

[8] [8]

Csajb´ ok, G

B. Csajb´ ok, G. Marino, F. Zullo: New maximum scattered l inear sets of the projective line, Finite Fields Appl. 54 (2018), 133–1 50. 19

work page 2018