Shunqi Lu

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Finite geometry

Preprint

Completing the classification of maximum scattered linear sets in \(\mathrm{PG}(1, q^5)\)

Shunqi Lu

Type
Preprint
Date
Version
v1 · Zenodo
DOI
10.5281/zenodo.22232593
Code
GitHub repository

Abstract

We complete the classification of maximum scattered \(\mathbb{F}_q\)-linear sets in \(\mathrm{PG}(1, q^5)\) up to \(\mathrm{P}\Gamma\mathrm{L}(2, q^5)\)-equivalence. Lia, Longobardi and Zanella reduced the problem to the pseudoregulus and LP-type families together with two residual candidate families, denoted (C3) and (C4), previously excluded only for \(q \le 25\). We prove that neither residual family contains a maximum scattered linear set for any prime power \(q\). Consequently, every maximum scattered \(\mathbb{F}_q\)-linear set in \(\mathrm{PG}(1, q^5)\) is equivalent to a member of the pseudoregulus or LP-type family.

The proof encodes collisions as rational points on linear sections of \(\mathrm{Gr}(2, 5)\), uses a finite-extension point-count identity to pass to dual collision sections, and analyzes the resulting degree-five Pfaffian curves through their Frobenius orbits and tangent lines.

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Keywords

  • Maximum scattered linear set
  • Linearized polynomial
  • Rank-metric code
  • Grassmannian
  • Pfaffian curve

MSC 2020 — 51E20 · 11T06 · 14M15 · 94B27

Key contributions

  1. The classification is completed Theorem 1.1

    For every prime power \(q\), every maximum scattered \(\mathbb{F}_q\)-linear set in \(\mathrm{PG}(1, q^5)\) is \(\mathrm{P}\Gamma\mathrm{L}(2, q^5)\)-equivalent to \(L_{x^q}\) or to \(L_{x^{q^s} + \delta x^{q^{5-s}}}\) with \(s \in \{1, 2\}\) and \(N(\delta) \notin \{0, 1\}\): the pseudoregulus and LP-type families. No member of either residual family, (C3) or (C4), is maximum scattered.

  2. An MRD code is shown nonextendable Corollary 1.2

    If \(U_\delta = \{(x, x^q + \delta x^{q^4}) : \mathrm{Tr}(x) = 0\}\) is scattered, then it is maximal with respect to inclusion among scattered \(\mathbb{F}_q\)-subspaces of \(K^2\); equivalently, its associated \([4, 2, 3]_{q^5/q}\) MRD code is nonextendable.

  3. A primal–dual point-count identity Section 2

    Collisions are encoded as rational points on a linear section \(\mathcal{C}\) of \(\mathrm{Gr}(2, 5)\), and a rank-stratified incidence count gives \(\#\mathcal{C}(E) = \#\mathcal{D}(E)\) over every finite extension \(E/\mathbb{F}_q\). The direct proof also covers characteristic two and sections without smoothness or expected-dimension hypotheses.

  4. A Pfaffian tangent criterion Section 3

    In each collision model the dual section is a height-three Pfaffian curve of degree five and arithmetic genus one. For a point-free section, the component through each point of a distinguished Frobenius orbit is the corresponding tangent line, and evaluation on those lines yields the reductions of (C3) and (C4).

Cite this preprint

BibTeX

@misc{lu2026scattered,
  author    = {Shunqi Lu},
  title     = {Completing the classification of maximum scattered
               linear sets in $\mathrm{PG}(1,q^5)$},
  year      = {2026},
  version   = {v1},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.22232593},
  url       = {https://doi.org/10.5281/zenodo.22232593}
}

Shunqi Lu. (2026). Completing the classification of maximum scattered linear sets in PG(1, q^5) (v1). Zenodo. https://doi.org/10.5281/zenodo.22232593