2021/05/10 by Liming Ma, Chaoping Xing, Ma, Liming +1
Computer Science · Engineering · #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2105.04370
openalex publication_date 2021/05/10 · openalex created_date 2021/05/24 · openalex updated_date 2026/07/28
The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to improve the Hasse-Weil bound for Artin-Schreier curves. In this paper, we focus on the number of rational places of the Artin-Schreier curve defined by yp-y=f(x) over the finite field \mathbbFq of characteristic p, where f(x) is a polynomial in \mathbbFq[x]. Our road map for attacking this problem works as follows. We first show that the function field Ef:=\mathbbFq(x,y) of the Artin-Schreier curve yp-y=f(x) is a subfield of some cyclotomic function field. We then make use of the class field theory to prove that the number of points of the curve is upper bounded by a function of a minimum distance of a linear code. By analyzing the minimum distance of this linear code, we can improve the Hasse-Weil bound and Serre bound for Artin-Schreier curves.