The set of pure gaps at several rational places in function fields

IF 1.2 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2024-12-28 DOI:10.1007/s10623-024-01556-x

Alonso S. Castellanos, Erik A. R. Mendoza, Guilherme Tizziotti

引用次数: 0

Abstract

In this work, we explore the use of maximal elements in generalized Weierstrass semigroups and their relationship with pure gaps, extending the results in Castellanos et al. [J Pure Appl Algebra 228(4):107513, 2024]. We provide a method to completely determine the set of pure gaps at several rational places in a function field F over a finite field, where the periods of certain places are the same, and determine its cardinality. As an example, we calculate the cardinality and provide a simple, explicit description of the set of pure gaps at several rational places distinct from the infinity place on Kummer extensions, offering a different characterization from that presented by Hu and Yang [Des Codes Cryptogr 86(1):211–230, 2018]. Furthermore, we present some applications in coding theory and AG codes with good parameters.

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函数域中若干有理位上的纯间隙集

本文研究了广义Weierstrass半群中极大元的使用及其与纯间隙的关系，推广了Castellanos等人的研究结果[J].应用数学学报，28(4):513 - 513。给出了在有限域上函数域F上若干有理点上纯间隙集的完全确定方法，其中某些点的周期相同，并确定了其基数。作为一个例子，我们计算了基数，并提供了一个简单的，明确的描述在几个有理性的地方与Kummer扩展上的无穷位不同的纯间隙集，提供了一个与Hu和Yang提出的不同的表征[Des Codes Cryptogr 86(1):211 - 230,2018]。此外，我们还介绍了它在编码理论和具有良好参数的AG码中的一些应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.