Partially scattered linearized polynomials and rank metric codes

TL;DR

This paper introduces two new generalizations of scattered linearized polynomials, explores their properties, and investigates their connections to linear sets and rank metric codes, expanding the theoretical framework in finite field coding theory.

Contribution

It defines L-$q^t$- and R-$q^t$-partially scattered polynomials, broadening the concept of scattered polynomials and linking them to linear sets and rank metric codes.

Findings

Defined L-$q^t$- and R-$q^t$-partially scattered polynomials

Established connections to linear sets and rank metric codes

Characterized the relationship with classical scattered polynomials

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.