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On the equational graphs over finite fields
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2020-03-13 , DOI: 10.1016/j.ffa.2020.101667
Bernard Mans , Min Sha , Jeffrey Smith , Daniel Sutantyo

In this paper, we generalize the notion of functional graph. Specifically, given an equation E(X,Y)=0 with variables X and Y over a finite field Fq of odd characteristic, we define a digraph by choosing the elements in Fq as vertices and drawing an edge from x to y if and only if E(x,y)=0. We call this graph as equational graph. In this paper, we study the equational graph when choosing E(X,Y)=(Y2f(X))(λY2f(X)) with f(X) a polynomial over Fq and λ a non-square element in Fq. We show that if f is a permutation polynomial over Fq, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.



中文翻译:

关于有限域上的方程图

在本文中,我们概括了功能图的概念。具体来说,给定一个方程ËXÿ=0在有限域上具有变量XYFq 的奇数特征,我们通过选择元素来定义有向图 Fq当且仅当作为顶点并从xy绘制边线时ËXÿ=0。我们将此图称为方程图。在本文中,我们研究选择时的方程图ËXÿ=ÿ2-FXλÿ2-FXFX 多项式 FqλFq。我们证明如果f是一个置换多项式Fq,则图的每个连接的分量都有一个哈密顿循环。此外,这些哈密顿循环可用于构造平衡二值序列。通过对低次排列的多项式f进行计算,似乎几乎所有这些图都是强连通的,并且如果该图是连通的,则在该图中会有许多哈密顿循环。

更新日期:2020-03-13
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