Let $f$ be the $mathbb{F}_q$-linear map over $mathbb{F}_{q^{2n}}$ defined by $xmapsto x+ax^{q^s}+bx^{q^{n+s}}$ 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. $ngeq5$ 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.
On certain linearized polynomials with high degree and kernel of small dimension
Olga Polverino;Ferdinando Zullo;
2021
Abstract
Let $f$ be the $mathbb{F}_q$-linear map over $mathbb{F}_{q^{2n}}$ defined by $xmapsto x+ax^{q^s}+bx^{q^{n+s}}$ 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. $ngeq5$ 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
