论文标题
在某些线性化的多项式上，高尺寸和小核的内核
On certain linearized polynomials with high degree and kernel of small dimension
论文作者
论文摘要
令$ f $为$ \ mathbb {f} _q $ -linear映射，$ \ mathbb {f} _ {q^{2n}} $由$ x \ mapsto x+ax+ax+ax^{q^s}+bx^s}+bx^{q^{q^{q^{q^{n+s}} $ with $ \ gcd（n，s）众所周知，$ f $的内核最多具有$ 2 $，正如CSAJBók等人所证明的那样。在“新的MRD代码家族”（2018年）中。对于$ n $足够大，例如$ n \ geq5 $当$ s = 1 $时，我们将$ b/a $的值分类为$ f $的内核最多具有$ 1 $。为此，我们将问题转化为对$ f $的一些代数曲线的研究；这允许将交叉理论和功能场理论与Hasse-Weil结合一起使用。我们的结果暗示了某些高度分散二项式的非分散性结果，以及对等级度量代码家族的渐近分类。
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 with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.
