Algebraic lattices coming from $${\mathbb {Z}}$$ -modules generalizing ramified prime ideals in odd prime degree cyclic number fields

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Applicable Algebra in Engineering Communication and Computing Pub Date : 2024-07-08 DOI:10.1007/s00200-024-00666-2

Antonio Aparecido de Andrade, Robson Ricardo de Araujo, Trajano Pires da Nobrega Neto, Jefferson Luiz Rocha Bastos

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引用次数: 0

Abstract

Lattice theory has shown to be useful in information theory, and rotated lattices with high modulation diversity have been extensively studied as an alternative approach for transmission over a Rayleigh fading channel, where the performance depends on the minimum product distance to achieve coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminants. With the construction of full-diversity algebraic lattices as our goal, in this work we present and study constructions of full-diversity algebraic lattices in odd prime dimensional Euclidean spaces from families of modules in cyclic number fields. These families include all the ramified prime ideals in each of these number fields. As immediate applications of our results, we present algebraic constructions from the densest lattices in dimensions 3 and 5.

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来自奇素数域中{${mathbb {Z}}$ -模块泛化夯素理想的代数网格

格子理论在信息论中非常有用，具有高调制分集的旋转格子已被广泛研究，作为在瑞利衰减信道上传输的另一种方法，其性能取决于实现编码增益的最小乘积距离。当我们使用完全实数字段时，旋转晶格的最大分集就能得到保证，而通过考虑具有最小判别式的字段，最小乘积距离就能得到优化。以构建全多样性代数网格为目标，我们在本研究中介绍并研究了奇素数维欧几里得空间中由循环数域中的模块族构建的全多样性代数网格。这些族包括每个数域中的所有夯素理想。作为我们结果的直接应用，我们提出了维数 3 和维数 5 中最密网格的代数构造。

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

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.