On certain linearized polynomials with high degree and kernel of small dimension

TL;DR

This paper classifies conditions under which certain high-degree linearized polynomials over finite fields have small kernels, using algebraic geometry tools, and explores implications for rank metric codes and scattered binomials.

Contribution

It provides a classification of parameters for high-degree linearized polynomials with small kernels, linking algebraic curves and intersection theory to finite field polynomial analysis.

Findings

Classified parameters for kernel dimension at most 1 when n ≥ 5 and s=1.

Connected polynomial kernel properties to algebraic curves and intersection theory.

Implications for non-scatteredness of high degree scattered binomials and rank metric codes.

Abstract

Let $f$ be the $\mathbb{F}_q$-linear map over $\mathbb{F}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajb\'ok et al. in "A new family of MRD-codes" (2018). For $n$ big enough, e.g. $n\geq5$ when $s=1$, we classify the values of $b/a$ such that the kernel of $f$ has dimension at most $1$. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of $f$; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.