Let f be the Fq-linear map over Fqjavax.xml.bind.JAXBElement@3e829f17 defined by x↦x+axqjavax.xml.bind.JAXBElement@df90dc4+bxqjavax.xml.bind.JAXBElement@5c033a86 with gcd⁡(n,s)=1. It is known that the kernel of f has dimension at most 2, as proved by Csajbók 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.

On certain linearized polynomials with high degree and kernel of small dimension / Polverino, O.; Zini, G.; Zullo, F.. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 225:2(2021), pp. 1-16. [10.1016/j.jpaa.2020.106491]

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

Zini G.
;
2021

Abstract

Let f be the Fq-linear map over Fqjavax.xml.bind.JAXBElement@3e829f17 defined by x↦x+axqjavax.xml.bind.JAXBElement@df90dc4+bxqjavax.xml.bind.JAXBElement@5c033a86 with gcd⁡(n,s)=1. It is known that the kernel of f has dimension at most 2, as proved by Csajbók 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.
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On certain linearized polynomials with high degree and kernel of small dimension / Polverino, O.; Zini, G.; Zullo, F.. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 225:2(2021), pp. 1-16. [10.1016/j.jpaa.2020.106491]
Polverino, O.; Zini, G.; Zullo, F.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/1258222
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