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On certain linearized polynomials with high degree and kernel of small dimension
Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.jpaa.2020.106491
Olga Polverino , Giovanni Zini , Ferdinando Zullo

Abstract Let f be the F q -linear map over F q 2 n defined by x ↦ x + a x q s + b x 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. 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.

中文翻译:

关于某些高阶小维核的线性化多项式

摘要 令 f 是 F q 2 n 上的 F q 线性映射,由 x ↦ x + axqs + bxqn + s 定义,gcd ⁡ ( n , s ) = 1 。正如 Csajbok 等人所证明的,已知 f 的核的维度最多为 2。在“一个新的 MRD 代码家族”(2018 年)中。对于足够大的 n,例如当 s = 1 时 n ≥ 5,我们对 b / a 的值进行分类,使得 f 的核的维度最多为 1。为此,我们将问题转化为一些代数曲线的研究相对于 f 的度数较小;这允许将交集理论和函数场理论与 Hasse-Weil 界一起使用。我们的结果意味着某些高度分散的二项式的非分散性结果,以及秩度量代码族的渐近分类。
更新日期:2021-02-01
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