有限域上加性码的基础

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-02-19 DOI:10.1016/j.ffa.2025.102592

Dipak K. Bhunia , Steven T. Dougherty , Cristina Fernández-Córdoba , Mercè Villanueva

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引用次数: 0

摘要

加性码最初是由Delsarte于1973年在关联方案的背景下引入的，最近由于它们在构造量子纠错码中的应用而引起了人们的兴趣。给出了元素来自有限域的加性码的基本结果，并利用群特征定义了正交关系。我们为这些加性码引入了一种类型，并探讨了生成集的独立性概念。此外，我们还根据加性代码的类型给出了其生成器矩阵的定义。我们还把加性码的类型与其正交码的类型联系起来。我们研究了与这些加性码相关的核族和秩。我们将加性码的等价性与其类型、核族和秩族以及对偶性联系起来。我们看到这些关系在加性码的分类中是如何起作用的。最后，给出了一种通用的编码和解码方法。

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Foundations of additive codes over finite fields

Additive codes were initially introduced by Delsarte in 1973 within the context of association schemes and recently they have become of interest due to their application in constructing quantum error-correcting codes.

We give foundational results for additive codes where the elements are from a finite field, and define the orthogonality relation using group characters. We introduce a type for these additive codes and explore the notion of independence for a generating set. Additionally, we provide a definition for a generator matrix of an additive code based on its type. We also relate the type of an additive code to the type of its orthogonal. We study a family of kernels and ranks associated with these additive codes. We relate the equivalence of additive codes to their type, the family of kernels and ranks, and duality. We see how these relations contribute in the classification of additive codes. Finally, we provide a general encoding and decoding method for these codes.

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.