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- W3043695256 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 Csajbok 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." @default.
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- W3043695256 date "2020-04-22" @default.
- W3043695256 modified "2023-09-23" @default.
- W3043695256 title "On certain linearized polynomials with high degree and kernel of small dimension" @default.
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