Completing the classification of maximum scattered linear sets in PG(1, q⁵)

Abstract

We complete the classification of maximum scattered 𝔽_q-linear sets in PG(1, q⁵) up to PΓL(2, q⁵)-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 ≤ 25. We prove that neither residual family contains a maximum scattered linear set for any prime power q. Consequently, every maximum scattered 𝔽_q-linear set in PG(1, q⁵) is equivalent to a member of the pseudoregulus or LP-type family.

The proof encodes collisions as rational points on linear sections of 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.