  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.
