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

Lia, Stefano; Longobardi, Giovanni; Zanella, Corrado

doi:10.5802/alco.483

Lia, Stefano; Longobardi, Giovanni; Zanella, Corrado

Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 327-355

Abstract

Every maximum scattered linear set in $\mathrm{PG}(1,q^5)$ is the projection of an $\mathbb{F}_q$-subgeometry $\Sigma $ of $\mathrm{PG}(4,q^5)$ from a plane $\Gamma $ external to the secant variety to $\Sigma $ [20]. The pair $(\Gamma ,\Sigma )$ will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in $\mathrm{PG}(1,q^5)$, namely those of pseudoregulus and LP type, have been characterized in the literature [6, 30]. Let $(\Gamma ,\Sigma )$ be a projecting configuration for a maximum scattered linear set in $\mathrm{PG}(1,q^5)$. Let $\sigma $ be a generator of $\mathbb{G}=\mathrm{P\Gamma L}(5,q^5)_\Sigma $, and $A=\Gamma \cap \Gamma ^{\sigma ^4}$, $B=\Gamma \cap \Gamma ^{\sigma ^3}$. If $A$ and $B$ are not both points, then the projected linear set is of pseudoregulus type [6]. Suppose that they are points. The rank of a point $X$ is the vectorial dimension of the span of the orbit of $X$ under the action of $\mathbb{G}$. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points $A$ and $B$ has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that $\mathrm{rk}\, A=\mathrm{rk}\, B=4$. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for $q\le 25$ no new maximum scattered linear set exists.

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Received: 2024-07-31
Revised: 2025-12-11
Accepted: 2026-01-15
Published online: 2026-04-30

DOI: 10.5802/alco.483

Classification: 51E20, 05B25
Keywords: finite projective space, linear set, linearized polynomial, scattered polynomial

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Lia, Stefano; Longobardi, Giovanni; Zanella, Corrado. Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$. Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 327-355. doi: 10.5802/alco.483

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