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

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article

Polverino, Olga¹  Zini, Giovanni¹  Zullo, Ferdinando¹ 

[1] Univ Campania Luigi Vanvitelli, Dipartimento Matemat & Fis, Viale Lincoln 5, I-81100 Caserta, Italy

关键词: Linearized polynomial;    Algebraic curve;    Linear set;    MRD code;    Hasse-Weil bound;   

DOI  :  10.1016/j.jpaa.2020.106491

来源: Elsevier

【 摘 要 】

Let f be the F-q-linear map over F-q(2n) defined by x bar right arrow x ax(q)(s) + bx(qn+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 [9]. 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. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free