Generalizing a family of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

Alessandro Giannoni; Giovanni Giuseppe Grimaldi; Giovanni Longobardi; Marco Timpanella

REVIEW 2 major objections 4 minor 23 references

Generalizing a family of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A quadrinomial family over F_{q^{2t}} is shown scattered under conditions strictly broader than previous results, yielding new MRD codes.

desk verdict A solid generalization of the known scattered quadrinomial family, with the caveat that a load-bearing decomposition lemma is asserted without proof for the new N(h)=1 range. read the letter →

arxiv 2601.09415 v4 pith:SPNGIHXO submitted 2026-01-14 math.CO

Alessandro Giannoni , Giovanni Giuseppe Grimaldi , Giovanni Longobardi , Marco Timpanella This is my paper · ORCID

classification math.CO MSC 11T0611T7194B05

keywords LinearizedpolynomialsScatteredMRDcodesFinitefieldsLinearsetsQuadrinomialsTrace-kernelRankmetric

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 establishes sufficient conditions under which the four-term $q^s$-linearized polynomial $\psi_{m,h,s}(X) = m(X^{q^s} - h^{1-q^{s(t+1)}}X^{q^{s(t+1)}}) + X^{q^{s(t-1)}} + h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}$ is a scattered polynomial over $F_{q^{2t}}$. Scatteredness means that the ratio $\psi(x)/x$ takes each value on at most $q$ inputs, which makes the associated rank-metric code $\{aX + b\psi\}$ a maximum-rank-distance (MRD) code. The theorem splits according to the parity of $t$ and the residue of $q$ modulo 4, with conditions on $m$ lying in $F_{q^t}$ and avoiding the sets $P_s^+$ and $P_s^-$ of $(q^s+1)$-st and $(q^s-1)$-st powers of trace-zero elements, and on the norm $N_{q^{2t}/q^t}(h)$. These conditions strictly include all previously published scatteredness results for this family. A reader who accepts the proof gets a new supply of scattered polynomials and MRD codes beyond the three known infinite families.

What carries the argument

The proof is carried by the decomposition of $F_{q^{2t}}$ as an $F_{q^t}$-vector space into the direct sum $\ker L_m \oplus \ker M$ (and $\operatorname{im} L_m \oplus \operatorname{im} M$), where $L_m = m(X^{q^s} - h^{1-q^{s(t+1)}}X^{q^{s(t+1)}})$ and $M = X^{q^{s(t-1)}} + h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}$. Two auxiliary $F_{q^t}$-linear maps $R = X^{q^{st}} + h^{q^{s(t-1)}-q^s}X$ and $T = X^{q^{st}} + h^{q^s-q^{s(t-1)}}X$ provide 1-dimensional kernels whose nonzero elements $\rho$ and $\tau = h^{q^{s(t-1)}-q^s}\rho$ give bases $\{1,\rho\}$ and $\{1,\tau\}$ of $F_{q^{2t}}$. Writing $\gamma$ in these bases turns the scatteredness test $\psi(\gamma x) = \gamma \psi(x)$ into a pair of equations whose coefficient contradictions force $m$ into the forbidden sets $P_s^\pm$.

What would settle it

A direct computer search for $q=3$, $t=3$, $s=1$: choose $h \in F_{3^6} \setminus F_{3^3}$ with $N_{3^6/3^3}(h)=1$ and $h^2 \neq -1$, choose $m \in F_{3^3} \setminus (P_1^+ \cup P_1^-)$, and compute $\dim_{F_{3^3}}(\ker L_m \cap \ker M)$. If the dimension is positive, Proposition 2.4 is false for the norm-one case; alternatively, test the scatteredness ratio directly by searching for two $F_3$-linearly independent $x,y \in F_{3^6}$ with $\psi_{m,h,s}(x)/x = \psi_{m,h,s}(y)/y$, which would disprove Theorem 3.4.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for $t \geq 3$, $q$ odd, $\gcd(s,2t)=1$, $\psi_{m,h,s}$ is scattered when (i) $t$ even, or $t$ odd with $q \equiv 1 \pmod{4}$, $m \notin P_s^+ \cup P_s^-$ and $N(h)=\pm 1$; or (ii) $t$ odd, $q \equiv 3 \pmod{4}$, and either $m \in P_s^+$ with $N(h)=-1$, or $m \notin P_s^+ \cup P_s^-$ with $N(h)=1$ and $h^2 \neq -1$. The proof writes $\psi = L_m + M$ and decomposes $F_{q^{2t}}$ as an $F_{q^t}$-vector space into direct sums of kernels and images of these summands, then forces the scattered condition $\psi(\gamma x) = \gamma \psi(x)$ through a $2 \times 2$ linear system whose possible solutions contradict the assumptions on $m$. For $t \geq 5$ the paper also proves the quadrinomials are $\Gamma L$-equivalent neither to pseudoregulus-type monomials nor to Lunardon–Polverino binomials, and it computes the stabilize

Load-bearing premise

The whole proof rests on Proposition 2.4, which asserts that $F_{q^{2t}}$ is the $F_{q^t}$-direct sum of the kernels (and of the images) of $L_m$ and $M$; the paper explicitly notes that for the norm-one, $h^2 \neq -1$ case this proposition is carried over from a companion paper by analogy, with the proof omitted, so if that extension fails, the central theorem has no foundation.

Editorial extensions

If this is right

For every triple (m,h,s) satisfying Theorem 1.1, the code {aX + bψ_{m,h,s} : a,b ∈ F_{q^{2t}}} is a linear MRD code with parameters (2t, 2t, q; 2t-1).
The families of pseudoregulus-type monomials and Lunardon–Polverino binomials are not ΓL-equivalent to ψ_{m,h,s} for t≥5, so the newly scattered polynomials are genuinely new objects.
The necessary-condition theorem shows that m∈P_s^- together with h∈F_{q^t} and N(h)=±1 forces ψ_{m,h,s} not to be scattered, settling one direction of the proposed characterization.
Since scatteredness is preserved under adjoints, the result also yields scattered adjoint polynomials and their associated linear sets.
The stabilizer computation fixes the right idealizer of the associated MRD code as F_{q^2}, an invariant that can distinguish these codes from others.

Reading between the lines

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

Conjecture 6.3 asserts the converse; if true, the sufficient conditions in Theorem 1.1 are exactly the scatteredness boundary, suggesting a complete classification of scattered quadrinomials of this shape.
Because P_s^± are independent of s (Lemma 2.2), the conditions may depend only on q and t and on whether m is a (q±1)-st power of a trace-zero element; this could be checked by testing the same property for s=1 in small examples.
A testable extension is to relax gcd(s,2t)=1 or allow q even; the proof's use of parity suggests the even-q case may need separate tools, but if the same direct-sum decomposition holds there, the method would generalize.
Theorem 4.3 gives explicit necessary conditions for two quadrinomials to be GL-equivalent; counting the number of inequivalent pairs (m,h) satisfying Theorem 1.1 would quantify how much larger this family is than previously known ones.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Dear colleague,

The thing to know: this is a genuine extension of the only known infinite family of maximum scattered quadrinomials, and the main theorem is coherent. The caveat is that one load-bearing lemma, Proposition 2.4, is asserted for the new N(h)=1, h^2≠-1 case without proof, and the proof of the main theorem leans on it.

What is actually new: Theorem 1.1 covers m outside P± under norm ±1 and adds the P+ case with norm −1; it strictly contains the subcases from [1,12,13,18,22,23]. The machinery is a long case analysis, but the structure is clear: split F_{q^{2t}} via kernels/images of the two summands, reduce the scatteredness equation to systems, then use norm conditions. Theorem 4.3 and Corollary 4.4 give GL-nonequivalence criteria and actually produce new examples; the stabilizer computation in Section 5 is also useful. This is solid subfield work, not a paradigm shift.

The soft spots. Proposition 2.4 is the foundation: the proofs of Theorems 3.2 and 3.4 start by writing every x as x1+x2 and then equating components in im L_m ⊕ im M. The paper says the N=1, h^2≠−1 case follows by similar techniques and omits it. Proposition 2.8 is in the same boat. That is a genuine gap, even though I think it is fillable: the stress-test note gives a plausible direct argument from (6), and the statement is probably true. But it is load-bearing, so the authors should supply the proof or cite a source that covers N=1. The Appendix Theorem 4.3 is long coefficient-comparison algebra; it is not machine-checked, and the exposition has at least one typo in Conjecture 6.3. These are minor relative to the main theorem. The citation practice is fine; leaning on prior lemmas from overlapping authors is not circular here because the target theorem is not assumed.

Who this is for: people working on scattered polynomials, MRD codes, and finite geometry. It deserves a serious referee. My recommendation: send it to peer review, with a clear request to prove Proposition 2.4 and Proposition 2.8 for N=1 (or cite a proof) before acceptance. If that lemma is fixed, the core result stands.

Referee Report

2 major / 4 minor

Summary. The paper studies F_q-scatteredness of the q^s-linearized quadrinomial ψ_{m,h,s} over F_{q^{2t}}, and states sufficient conditions in Theorem 1.1 depending on t, q mod 4, the norm N_{q^{2t}/q^t}(h), and membership of m in the sets P_s^+, P_s^-. The proof splits into the cases N(h)=-1 and N(h)=1: it writes ψ = L_m + M, uses a direct-sum decomposition of F_{q^{2t}} into kernels and images of L_m and M, and then performs a detailed case analysis on a putative multiplier γ. The paper also claims that these results strictly include previous scatteredness results for ψ_{m,h,s}; it proves non-equivalence with pseudoregulus-type and LP-type polynomials, gives a classification of GL-equivalences among ψ_{m,h,s} in Theorem 4.3, computes stabilizers, and proposes necessary conditions and a conjecture.

Significance. If Theorem 1.1 is correct, this is a genuine contribution: it unifies and extends the previously known scattered quadrinomial families, gives explicit new scattered polynomials, and therefore new MRD codes. The proof is largely a transparent, parameter-free case analysis, and the paper clearly identifies the new cases. The main weakness is that a load-bearing direct-sum lemma for the new N(h)=1 case is explicitly left unproved, and another used proposition is only partially proved. The result is plausible and likely fixable, but the manuscript in its present form does not fully support the central claim in the new branch.

major comments (2)

[Section 2, Proposition 2.4] The text before Proposition 2.4 says that results proved in [12] for N(h)=-1 also hold for N(h)=1, h^2≠-1, 'since the techniques are similar', and omits the proof. This decomposition is used at the first step of Theorems 3.2 and 3.4, and again after Eq. (11), to equate the im L_m and im M components and to conclude x=x1=x2=0 when a=0. Since the N(h)=1 branch is exactly the genuinely new part of Theorem 1.1, a citation to [12] cannot cover it. Please supply the proof, e.g. using (6) to show that a nonzero intersection of either the kernel pair or the image pair forces h^{q^{2s}+1}=-ε, contradicting Proposition 2.3 for both ε=±1. Propositions 2.6 and 2.7 are also covered by the same omitted-proof sentence and are used in deriving (11); include their proofs as well.
[Section 3.2, Proposition 2.8] Proposition 2.8 is stated for both N(h)=-1 and N(h)=1, h^2≠-1, but the proof computes only the N(h)=-1 case and ends with 'a similar argument applies' for N(h)=1. This is not a purely cosmetic variant: the norm condition changes the signs in (6), and Proposition 2.3(ii) must be invoked. The proposition is used in Theorem 3.4, Case 3.1, Eq. (33), to conclude m∈P_s^+ and obtain the contradiction. Please write out the N(h)=1 computation in full, explicitly showing the use of h^{q^{2s}+1}≠-1.

minor comments (4)

[Section 6, Conjecture 6.3] The polynomial displayed in Conjecture 6.3 is not the ψ_{m,h,s} of Eq. (3): the second term has exponent X^{q^{st}-1} and the third term is X^{q^{s(t+1)}}, so the displayed object is not even q^s-linearized. The conjecture should refer to the polynomial (3).
[Section 2, Lemma 2.5] Lemma 2.5 is stated without proof. It is elementary, but part (ii) is used in the proofs of Theorems 3.2 and 3.4, so a short verification should be included.
[Appendix, Theorem 4.3] The proof of Theorem 4.3 contains many compressed reductions, e.g. 'which in this case is equivalent to'. Since Theorem 4.3 supports Corollary 4.4 and the non-equivalence claims, expanding these reductions would make the argument substantially easier to verify.
[Throughout] There are minor typographical issues, e.g. 'the the' in Lemma 2.1(i). Also, Theorem 1.1 states (m,h)∈F_{q^t}×F_{q^{2t}}, while the definition (3) and the proof require m,h nonzero; this should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; the N(h)=1 branch is conditional on an unproved, self-cited extension of Prop. 2.4, but the theorem is not equivalent to its inputs.

full rationale

No fitted input is called a prediction, and no quantity is renamed as a derived result. The proof of Theorems 3.2 and 3.4 is a conditional derivation from the norm hypotheses N=±1 and the exclusion m∉P_s^+∪P_s^-; the sets P_s^± are defined by the same q^s±1 power conditions that appear naturally in the later contradictions, but the proof does not merely read off the conclusion from the definition. The main caveat is the passage before Proposition 2.4: 'Although the next results are proven in [12] for N_{q^{2t}/q^t}(h)=-1, these hold for N_{q^{2t}/q^t}(h)=1 and h^2≠-1, as well. Since the techniques are similar, we omit the proof.' This proposition supplies the direct-sum decomposition F_{q^{2t}}=ker L_m ⊕ ker M and im L_m ⊕ im M that is used at the first step of Theorem 3.4; Proposition 2.8, also justified for N=1 only by 'A similar argument applies', is used in Case 3.1 to force m∈P_s^+. Thus the genuinely new N(h)=1 branch rests on an unproved extension of lemmas from a paper with overlapping authors, which is a load-bearing proof gap. However, this is not circularity in the sense of the target theorem being assumed or of an equation reducing to its own input: [12] proves the N=-1 case, and the N=1 extension is an asserted analogue, not a reuse of the theorem being proved. The verdict is therefore conditional: if Prop. 2.4/2.8 are valid for N=1, the derivation is self-contained modulo standard lemmas; if not, Theorem 1.1(ii)(b) lacks a foundation. That is a correctness risk, not a circular derivation, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants; m,h,s,t,q are quantifiers. The load-bearing premises are finite-field decompositions and norm lemmas; one of them (Prop 2.4 for N(h)=1) is asserted without proof. P± are subsets of F_qt defined from the trace kernel, not new postulated objects.

assumptions (5)

standard math F_{q^{2t}} = F_{qt} ⊕ ker Tr_{q2t/qt}; products/odd powers in ker Tr behave as in Lemma 2.1
Used throughout Section 3 to split x=x1+x2 into kernel components.
ad hoc to paper For N(h)=±1, ker L_m ⊕ ker M = im L_m ⊕ im M = F_{q^{2t}} (Prop 2.4)
Asserted without proof for N(h)=1, h²≠−1; the case split of Theorems 3.2 and 3.4 depends on it.
standard math Norm conditions imply h^{q^{2s}+1}≠±1 and h^{q^{s(t−2)}}≠−h (Prop 2.3)
Cited/proven from [12]; used in each contradiction step.
standard math Intersection-number characterizations: int=1 for pseudoregulus, int=2 for LP-type
Used in Prop 4.1 to prove non-equivalence; imported from [5,23].
domain assumption q odd, t≥3, gcd(s,2t)=1
The theorem's stated domain; scatteredness outside this domain is not addressed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Generalizing a family of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$." pith.science (2026). https://pith.science/paper/SPNGIHXO

@misc{pith2026260109415,
  author       = {Pith},
  title        = {Pith review of: Generalizing a family of scattered quadrinomials in $\mathbbF_q^2t[X]$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPNGIHXO}},
  note         = {Machine review of arXiv:2601.09415}
}

abstract

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $\mathbb{F}_{q^n}[X]$ are known to exist for infinitely many values of $n$ and $q$: (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $\psi_{m,h,s}$, are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 1 linked inside Pith

[12]

Longobardi, G

G. Longobardi, G. Marino, R. Trombetti, Y. Zhou. A large family of maximum scattered linear sets of (1,q^n) and their associated MRD codes. Combinatorica 43, 681–716 (2023)

2023
[1]

Bartoli, C

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

2020
[2]

Blokhuis, M

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

2000
[3]

Casarino, G

V. Casarino, G. Longobardi, C. Zanella. Scattered linear sets in a finite projective line and translation planes, Linear Algebra Appl. 650, 286–298 (2022)

2022
[4]

Csajbók, G

B. Csajbók, G. Marino, O. Polverino, C. Zanella. A new family of MRD--codes. Linear Algebra Appl. 548, 203--220 (2018)

2018
[5]

Csajbók, C

B. Csajbók, C. Zanella. On scattered linear sets of pseudoregulus type in (1, q^t) , Finite Fields Appl. 41 (2016), 34–54

2016
[6]

Csajbók, C

B. Csajbók, C. Zanella. On the equivalence of linear sets. Des. Codes Cryptogr. , 81(2) (2016), 269–281

2016
[7]

Delsarte

P. Delsarte. Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory, Ser. A , 81, 226--241 (1978)

1978

Show all 23 references

[8]

R. Lidl, H. Niederreiter. Finite fields, 2nd edition, volume 20. Cambridge University Press (1997)

1997
[9]

Grimaldi, S

G.G. Grimaldi, S. Gupta, G. Longobardi, R. Trombetti A geometric characterization of known maximum scattered linear sets of (1, q^n) , arXiv:2405.01374

arXiv
[10]

Gupta, G

S. Gupta, G. Longobardi, R. Trombetti. On the equivalence issue of a class of 2-dimensional linear maximum rank distance codes. Linear Algebra Appl. , 678, 33--56 (2023)

2023
[11]

Longobardi

G. Longobardi. Scattered polynomials: an overview on their properties, connections and applications. Art of Discrete and Applied Mathematics , 1–25 (2024)

2024
[13]

Longobardi, C

G. Longobardi, C. Zanella. Linear sets and MRD-codes arising from a class of scattered linearized polynomials. J. Algebr. Combin. 53 , 639--661 (2021)

2021
[14]

Longobardi, C

G. Longobardi, C. Zanella. A standard form for scattered linearized polynomials and properties of the related translation planes. J. Algebr. Combin. 59, 917--937 (2024)

2024
[15]

Lunardon, O

G. Lunardon, O. Polverino. Blocking sets and derivable partial spreads. J. Algebr. Combin. 14 , 49--56 (2001)

2001
[16]

Lunardon, O

G. Lunardon, O. Polverino. Translation ovoids of orthogonal polar spaces. Forum Math. 16 , 663--669 (2004)

2004
[17]

Lunardon, R

G. Lunardon, R. Trombetti, Y. Zhou. On kernels and nuclei of rank metric codes. J. Algebr. Combin. 46, 313--340 (2017)

2017
[18]

A. Neri, P. Santonastaso, F. Zullo. Extending two families of maximum rank distance codes . Finite Fields Appl. 81 (2022)

2022
[19]

Polverino, F

O. Polverino, F. Zullo. Connections between scattered linear sets and MRD-codes. Bulletin of the ICA. 89, (2020)

2020
[20]

J. Sheekey. A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3) , 475--488 (2016)

2016
[21]

J. Sheekey. MRD codes: Constructions and connections. In K.-U. Schmidt and A. Winterhof (eds.), Combinatorics and Finite Fields. Difference Sets, Polynomials, Pseudorandomness and Applications , 255--286, De Gruyter (2019)

2019
[22]

Smaldore, C

V. Smaldore, C. Zanella, F. Zullo. New scattered quadrinomials. Linear Algebra Appl. 702, 143--160 (2024)

2024
[23]

Zanella, F

C. Zanella, F. Zullo. Vertex properties of maximum scattered linear sets of (1, q^n) . Discrete Math. 343(5) (2020)

2020

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [12]

Longobardi, G

G. Longobardi, G. Marino, R. Trombetti, Y. Zhou. A large family of maximum scattered linear sets of (1,q^n) and their associated MRD codes. Combinatorica 43, 681–716 (2023)

2023

[2] [1]

Bartoli, C

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

2020

[3] [2]

Blokhuis, M

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

2000

[4] [3]

Casarino, G

V. Casarino, G. Longobardi, C. Zanella. Scattered linear sets in a finite projective line and translation planes, Linear Algebra Appl. 650, 286–298 (2022)

2022

[5] [4]

Csajbók, G

B. Csajbók, G. Marino, O. Polverino, C. Zanella. A new family of MRD--codes. Linear Algebra Appl. 548, 203--220 (2018)

2018

[6] [5]

Csajbók, C

B. Csajbók, C. Zanella. On scattered linear sets of pseudoregulus type in (1, q^t) , Finite Fields Appl. 41 (2016), 34–54

2016

[7] [6]

Csajbók, C

B. Csajbók, C. Zanella. On the equivalence of linear sets. Des. Codes Cryptogr. , 81(2) (2016), 269–281

2016

[8] [7]

Delsarte

P. Delsarte. Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory, Ser. A , 81, 226--241 (1978)

1978