部分分散线性化多项式与秩度量码
Partially scattered linearized polynomials and rank metric codes
摘要：
如果对于任何$y,z\in\mathbb F_{q^n}$，条件$zf(y)-yf(z)=0$意味着$y$和$z$是$\mathbb F_{q}$线性相关的，则线性化多项式$f(x)\in\mathbb F_{q^n}[x]$被称为分散的。本文定义并研究了离散线性化多项式概念的两种推广。设$t$是$n$的一个非平凡正因子.通过削弱定义散射线性化多项式的性质，以这样的方式引入L- $q^t$ -部分散射和R- $q^t$ -部分散射线性化多项式，使得散射线性化多项式恰好是L- $q^t$ -和R- $q^t$ -部分散射的那些。此外，部分散布多项式，线性集和秩度量码之间的联系。
英文摘要：
A linearized polynomial $f(x)\in\mathbb F_{q^n}[x]$ is called scattered if for any $y,z\in\mathbb F_{q^n}$, the condition $zf(y)-yf(z)=0$ implies that $y$ and $z$ are $\mathbb F_{q}$-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are defined and investigated. Let $t$ be a nontrivial positive divisor of $n$. By weakening the property defining a scattered linearized polynomial, L-$q^t$-partially scattered and R-$q^t$-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-$q^t$- and R-$q^t$-partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited.
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