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

Abstract

Let $f$ be the ${\mathbb{F}}_q$-linear map over ${\mathbb{F}}_{q^{2n}}$ defined by $x\mapsto x+a{x}^{q^s}+b{x}^{q^{n+s}}$ with $\gcd (n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajbók et al. in "A new family of MRD-codes" (2018). For $n$ big enough, e.g. $n\ge 5$ 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.

References in corpus (6)

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• Scalar $q$-subresultants and Dickson matrices
• Identifiers for MRD-codes
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• Towards the full classification of exceptional scattered polynomials