Partially scattered linearized polynomials and rank metric codes

Abstract

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.