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].

线性化多项式因其在几何和代数领域的应用而引起了很多关注。设q为素数幂，n为正整数，σ为$\mathrm{加尔}({\mathbb{F}}_{q^n}：{\mathbb{F}}_q)$. 在本文中，我们提供的系数封闭公式σ -trinomial ˚F过${\mathbb{F}}_{q^n}$确保f的核的维数等于其σ度，即具有最大核的线性化多项式。因此，我们展示了具有最大核的线性化三项式的明确示例，并表征了具有3 次和 4 次σ 的那些。我们的技术依赖于 [24] 中开发的工具。最后，我们应用这些结果来研究 [8] 中引入的一类秩度量代码，以构造准子域多项式和循环子空间代码，获得对 [37] 中提出的猜想的新的显式构造。