论文标题
最大内核的线性化三项元素
Linearized trinomials with maximum kernel
论文作者
论文摘要
线性化的多项式引起了很多关注，因为它们在几何和代数区域都有应用。令$ q $为主要功率，$ n $为正整数，$σ$是$ \ mathrm {gal}的生成器（\ Mathbb {f} _ {q^n} \ colon \ colon \ mathbb {f} _q）$。在本文中，我们为$σ$ - 三项$ f $ f $ over $ \ mathbb {f} _ {q^n} $的系数提供了封闭式公式，这确保了$ f $的核心的尺寸等于其$σ$ - 程度，这是与最大值的polynomials与最大值的二级化值。结果，我们提出了具有最大内核的线性化三项元素的明确示例，并表征了那些具有$σ$ -Degree $ 3 $和$ 4 $的人。我们的技术取决于[24]中开发的工具。最后，我们将这些结果应用于[8]中引入的一类等级公制，以构建准赛赛多项式和循环子空间代码，从而为[37]中提出的猜想获得了新的显式结构。
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 $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. 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].
