Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic
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2023
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| _version_ | 1866917668921016320 |
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| author | Bartoli, Daniele Longobardi, Giovanni Marino, Giuseppe Timpanella, Marco |
| author_facet | Bartoli, Daniele Longobardi, Giovanni Marino, Giuseppe Timpanella, Marco |
| contents | In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajbók, G. Marino and F. Zullo: New maximum scattered linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G. Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial $x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$, Linear Algebra Appl. 591 (2020), 99-114], the authors proved that the trinomial $f_c(X)=X^{q}+X^{q^{3}}+cX^{q^{5}}$ of $\mathbb{F}_{q^6}[X]$ is scattered under the assumptions that $q$ is odd and $c^2+c=1$. They also explicitly observed that this is false when $q$ is even. In this paper, we provide a different set of conditions on $c$ for which this trinomial is scattered in the case of even $q$. Using tools of algebraic geometry in positive characteristic, we show that when $q$ is even and sufficiently large, there are roughly $q^3$ elements $c \in \mathbb{F}_{q^6}$ such that $f_{c}(X)$ is scattered. Also, we prove that the corresponding MRD-codes and $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_12829 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic Bartoli, Daniele Longobardi, Giovanni Marino, Giuseppe Timpanella, Marco Combinatorics 05B25, 06E30, 11T06, 51E20, 51E22 In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajbók, G. Marino and F. Zullo: New maximum scattered linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G. Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial $x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$, Linear Algebra Appl. 591 (2020), 99-114], the authors proved that the trinomial $f_c(X)=X^{q}+X^{q^{3}}+cX^{q^{5}}$ of $\mathbb{F}_{q^6}[X]$ is scattered under the assumptions that $q$ is odd and $c^2+c=1$. They also explicitly observed that this is false when $q$ is even. In this paper, we provide a different set of conditions on $c$ for which this trinomial is scattered in the case of even $q$. Using tools of algebraic geometry in positive characteristic, we show that when $q$ is even and sufficiently large, there are roughly $q^3$ elements $c \in \mathbb{F}_{q^6}$ such that $f_{c}(X)$ is scattered. Also, we prove that the corresponding MRD-codes and $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones. |
| title | Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic |
| topic | Combinatorics 05B25, 06E30, 11T06, 51E20, 51E22 |
| url | https://arxiv.org/abs/2307.12829 |