Linearized trinomials with maximum kernel

Some Results on Linearized Trinomials that Split Completely

Linearized polynomials over finite fields have been much studied over the last several decades. Recently there has been a renewed interest in linearized polynomials because of new connections to coding theory and finite geometry. We consider the problem of calculating the rank or nullity of a linearized polynomial $L(x)=\sum_{i=0}^{d}a_i x^{q^i}$ (where $a_i\in \mathbb{F}_{q^n}$) from the coefficients $a_i$. The rank and nullity of $L(x)$ are the rank and nullity of the associated $\mathbb{F}_q$-linear map $\mathbb{F}_{q^n} \longrightarrow \mathbb{F}_{q^n}$. McGuire and Sheekey defined a $d\ti…

具有显式逆的q线性化置换多项式的稀疏提升

Sparse Lifts of q-Linearized Permutation Polynomials with Explicit Inverses

该研究构造了保持复合逆的q线性化置换多项式的稀疏提升，得到二维二项式、三维三项式等的显式逆，为有限域上置换多项式的构造提供了新方法。

On certain linearized polynomials with high degree and kernel of small dimension

Let $f$ be the $\mathbb{F}_q$-linear map over $\mathbb{F}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajbók et al. in "A new family of MRD-codes" (2018). For $n$ big enough, e.g. $n\geq5$ when $s=1$, we classify the values of $b/a$ such that the kernel of $f$ has dimension at most $1$. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of $f$; this allows to use intersection theory and function field theory together w…