论文标题
有限字段上椭圆功能字段的自动形态组的组结构及其应用于最佳的本地维修代码
The group structures of automorphism groups of elliptic function fields over finite fields and their applications to optimal locally repairable codes
论文作者
论文摘要
椭圆形曲线在代数封闭场上的自动形态群是众所周知的。但是，对于编码理论和密码学中的各种应用，我们通常需要应用在有限领域定义的自动形态。尽管我们认为在社区中众所周知，在有限领域的椭圆形曲线的自动形态群体，但在文献中我们找不到这一点。然而，在本文中，我们显示了有限场上椭圆曲线的自动形态群的组结构。更重要的是，我们表征了有限场上椭圆曲线的自动形态组的亚组和亚伯群亚组。 尽管对该主题具有理论上的兴趣，但我们的研究在很大程度上是由最佳当地修复代码的构建的动机。在纸张\ cite {JMX20}中给出了使用自动形态功能字段的最佳局部维修代码的第一个研究，其中采用了射影线的自动形态组。在\ cite {mx19}中，这个想法进一步生成了椭圆曲线，其中仅使用了无限度的自动形态。由于椭圆曲线的$ 24 $自动形态固定了无限的点，因此该结构中最佳当地维修代码的本地性上限为$ 23 $。在有限场上研究椭圆曲线的自动形态组的亚组和阿贝尔亚组的主要动机之一是消除对区域的约束。
The automorphism group of an elliptic curve over an algebraically closed field is well known. However, for various applications in coding theory and cryptography, we usually need to apply automorphisms defined over a finite field. Although we believe that the automorphism group of an elliptic curve over a finite field is well known in the community, we could not find this in the literature. Nevertheless, in this paper we show the group structure of the automorphism group of an elliptic curve over a finite field. More importantly, we characterize subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field. Despite of theoretical interest on this topic, our research is largely motivated by constructions of optimal locally repairable codes. The first research to make use of automorphism group of function fields to construct optimal locally repairable codes was given in a paper \cite{JMX20} where automorphism group of a projective line was employed. The idea was further generated to an elliptic curve in \cite{MX19} where only automorphisms fixing the point at infinity were used. Because there are at most $24$ automorphisms of an elliptic curve fixing the point at infinity, the locality of optimal locally repairable codes from this construction is upper bounded by $23$. One of the main motivation to study subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field is to remove the constraints on locality.
