Existence of maximum scattered linear sets of types (C3) and (C4) in PG(1, q^5)

Determine whether there exist maximum scattered F_q-linear sets in PG(1, q^5) belonging to the classes labeled (C3) or (C4), namely the families defined by L_{η,ρ} = { (η(x − x^q) + Tr_{q^5/q}(ρx), x − x^q) : x ∈ F_{q^5}^* } with η, ρ ∈ F_{q^5}, η ≠ 0, Tr_{q^5/q}(η) = 0, and Tr_{q^5/q}(ρ) = 0, and by L = { (x, ξ(x + x^{q^2}) + x^{q^3} + x^{q^4}) : x ∈ F_{q^5}^* } with N_{q^5/q}(ξ) = 1. Ascertain whether these classes are non-empty by constructing explicit examples or prove that they are empty.

Background

The paper surveys known families of scattered polynomials and the associated maximum scattered linear sets, and it discusses classification results in PG(1, q^n) for small n. For n = 5, the authors outline four candidate types (C1)–(C4) for maximum scattered linear sets, where (C1) is pseudoregulus type and (C2) is Lunardon–Polverino (LP) type.

The remaining families (C3) and (C4) correspond to potential new types that would arise when the projecting vertex Γ in PG(4, q) has intersection points A = Γ ∩ Γ^σ and B = Γ ∩ Γ^{σ^2} both of rank 4. Computational evidence shows no examples for q ≤ 25, leaving open whether these classes actually contain maximum scattered linear sets at all. Resolving this would complete the classification for n = 5 and clarify the landscape of known constructions.