论文标题
支持3设计的无限循环和否定码家族
Infinite families of cyclic and negacyclic codes supporting 3-designs
论文作者
论文摘要
编码理论与组合$ t $ - 设计之间的相互作用是组合主义者和编码理论家多年来的热门话题。在过去的50年中，建造了一些无限循环代码的家庭，这些循环代码为$ 3 $ designs的家属建成。但是，文献中尚无一系列无限的否定码家族来支持无限的$ 3 $ designs。这是本文的主要动机。令$ q = p^m $，其中$ p $是一个奇数，$ m \ geq 2 $是整数。本文的目的是在$ \ gf（q）上介绍一个无限的环保代码家族，该家族支持一个无限的$ 3 $ designs家族和两个超过$ \ gf（q^2）$的无限型否定代码家族，支持两个无限的家族$ 3 $ designs。确定这些代码的参数和权重分布。研究了这些NegacyClic代码的子场子代码（Q）$。还提出了三个无限的MDS代码家族。本文还介绍了一本支持$ 4 $ - 设计和六个开放问题的GF（$ 4 $）上的Constacyclic代码（$ 4 $）。
Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of $3$-designs has been reported in the literature. This is the main motivation of this paper. Let $q=p^m$, where $p$ is an odd prime and $m \geq 2$ is an integer. The objective of this paper is to present an infinite family of cyclic codes over $\gf(q)$ supporting an infinite family of $3$-designs and two infinite families of negacyclic codes over $\gf(q^2)$ supporting two infinite families of $3$-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over $\gf(q)$ are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over GF($4$) supporting a $4$-design and six open problems are also presented in this paper.
