Lifting iso-dual algebraic geometry codes

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2024-05-07 DOI:10.1007/s10623-024-01412-y

María Chara, Ricardo Podestá, Luciane Quoos, Ricardo Toledano

引用次数: 0

Abstract

In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field \(\mathbb {F}_{q}\) with q elements. Given a finite separable extension \(\mathcal {M}/\mathcal {F}\) of function fields and an iso-dual AG-code \(\mathcal {C}\) defined over \(\mathcal {F}\), we provide a general method to lift the code \(\mathcal {C}\) to another iso-dual AG-code \(\tilde{\mathcal {C}}\) defined over \(\mathcal {M}\) under some assumptions on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian p-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the GGS function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.

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提升等二代数几何代码

在这项工作中，我们研究了在具有 q 个元素的有限域 \(\mathbb {F}_{q}\) 上生成等双代数几何（AG）代码的问题。给定函数域的有限可分离扩展 \(\mathcal {M}/\mathcal {F}\) 和定义在 \(\mathcal {F}\) 上的等双 AG 代码 \(\mathcal {C}\)、我们提供了一种一般方法，在对所涉及的不同指数的奇偶性做一些假设的情况下，将代码 \(\mathcal {C}\) 提升到定义在 \(\mathcal {M}\) 上的另一个等双 AG 代码 \(\tilde/{mathcal {C}\) 。我们应用这种方法把有理函数域上的等双 AG 代码提升到基本无住民 p 扩展，比如由赫尔墨斯、铃木和一个由 GGS 函数域覆盖的最大函数域定义的等双 AG 代码。我们还获得了定义在环函扩展上的长二元和三元等双 AG 代码。

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.