Linearized trinomials with maximum kernel

Linearized polynomials have attracted a lot of attention because their applications in both geometric and algebraic areas. Let q be prime power, n positive integer σ generator Gal(Fqn:Fq). In this paper we provide closed formulas for the coefficients σ-trinomial f over Fqn which ensure that dimension kernel equals its σ-degree, is linearized with maximum kernel. As consequence, present explicit examples trinomials characterize those having σ-degree 3 4. Our techniques rely on tools developed [24]. Finally, apply these results to investigate class rank metric codes introduced [8], construct quasi-subfield cyclic subspace codes, obtaining new constructions conjecture posed [37].