当前位置: X-MOL 学术Finite Fields Their Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Existence of primitive 2-normal elements in finite fields
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2021-04-29 , DOI: 10.1016/j.ffa.2021.101864
Josimar J.R. Aguirre , Victor G.L. Neumann

An element αFqn is normal over Fq if B={α,αq,αq2,,αqn1} forms a basis of Fqn as a vector space over Fq. It is well known that αFqn is normal over Fq if and only if gα(x)=αxn1+αqxn2++αqn2x+αqn1 and xn1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al. (2013): an element αFqn is k-normal over Fq if the greatest common divisor of the polynomials gα[x] and xn1 in Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal.

Huczynska et al. made the question about the pairs (n,k) for which there exist primitive k-normal elements in Fqn over Fq and they got a partial result for the case k=1, and later Reis and Thomson (2018) completed this case. The Primitive Normal Basis Theorem solves the case k=0. In this paper, we solve completely the case k=2 using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of k-normal elements in Fqn.



中文翻译:

有限域中原始2法线元素的存在

一个元素 αFqñ 是正常的 Fq 如果 ={ααqαq2个αqñ-1个} 构成 Fqñ 作为向量空间 Fq。众所周知αFqñ 是正常的 Fq 当且仅当 GαX=αXñ-1个+αqXñ-2个++αqñ-2个X+αqñ-1个Xñ-1个 相对来说比较重要 Fqñ,即他们最大公约数的程度 Fqñ[X]使用等价,在Huczynska等人中引入了k正态元素的概念。(2013):元素αFqñk -normal overFq 如果多项式的最大公约数 Gα[X]Xñ-1个Fqñ[X]具有度k ; 因此通常意义上正常的元素为0-正常。

Huczynska等。提出了关于两人的问题ñķ为此,在其中存在原始k-法线元素Fqñ 超过 Fq 他们得到了部分结果 ķ=1个,后来Reis和Thomson(2018)完成了本案。原始正态基础定理解决了这种情况ķ=0。在本文中,我们完全解决了这种情况ķ=2个使用高斯和的估计值和计算机的使用,我们还获得了存在k-法线元素的新条件Fqñ

更新日期:2021-04-29
down
wechat
bug