Linearized trinomials with maximum kernel

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