On certain linearized polynomials with high degree and kernel of small dimension (Q2196349)

Let \(f_{a,b,s}(x)= x + ax^{q^s} + bx^{q^{n+s}}\) be an \({\mathbb F}_q\)-linear map on \({\mathbb F}_{q^{2n}}\), with \(\gcd(n, s) = 1\). It is known that the kernel of \(f_{a,b,s}\) has dimension at most 2 [\textit{B. Csajbók} et al., Linear Algebra Appl. 548, 203--220 (2018; Zbl 1401.51006)]. The authors prove that if \(n\) is large enough then for every \(\delta \in {\mathbb F}_{q^{2n}}^*\) with \(\mathrm{N}_{q^{2n}/q^n} (\delta) \ne 1\) there exists \(a \in {\mathbb F}_{q^{2n}}^*\) such that \(\dim_{\mathbb F_q} \ker (f_{a,\delta a,s}(x)) = 2\), where \(\mathrm{N}_{q^{2n}/q^n} (x) = x^{q+1}\). Applications of this result to scattered polynomials, linear sets and rank metric codes are considered.