Homogenization of binary linear codes and their applications

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-02-06 DOI:10.1016/j.ffa.2025.102589

Jong Yoon Hyun , Nilay Kumar Mondal , Yoonjin Lee

引用次数: 0

Abstract

We introduce a new technique, called homogenization, for a systematic construction of augmented codes of binary linear codes, using the defining set approach in connection to multi-variable functions. We explicitly determine the parameters and the weight distribution of the homogenized codes when the defining set is either a simplicial complex generated by any finite number of elements, or the difference of two simplicial complexes, each of which is generated by a single maximal element. Using this homogenization technique, we produce several infinite families of optimal codes, self-orthogonal codes, minimal codes, and self-complementary codes. As applications, we obtain some best known quantum error-correcting codes, infinite families of intersecting codes (used in the construction of covering arrays), and we compute the Trellis complexity (required for decoding) for several families of codes as well.

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二进制线性码的均匀化及其应用

我们引入了一种新的技术，称为均匀化，用于二元线性码的增广码的系统构造，使用定义集方法连接到多变量函数。当定义集是由任意有限个元素生成的简单复合体，或由单个极大元素生成的两个简单复合体之差时，我们明确地确定了均质码的参数和权分布。利用这种均匀化技术，我们得到了几个无限族的最优码、自正交码、最小码和自互补码。作为应用，我们获得了一些最著名的量子纠错码，无限族的相交码（用于覆盖阵列的构造），我们也计算了几个族码的网格复杂度（解码所需）。

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

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.