论文标题
部分分散的线性化多项式和等级度量代码
Partially scattered linearized polynomials and rank metric codes
论文作者
论文摘要
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.在本文中，定义和研究了散射线性多项式概念的两个概括。令$ t $为$ n $的非平凡的正分离。通过削弱定义分散的线性多项式的属性，l- $ q^t $逐渐分散和r- $ q^t $ - 通过这种方式引入了散布的线性化的多项式的方式，以使得均精确地是l- $ q^t $ - q^t $ - q^t $ q^t $ - Q^t $ -sparted-parted spacked。同样，展示了部分分散的多项式，线性集和等级度量代码之间的连接。
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.
