On certain linearized polynomials with high degree and kernel of small dimension
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Let f be the F_q-linear map over F_q^2n defined by x x+ax^q^s+bx^q^n+s with (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≥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.
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