Title: The Subfield Codes of Some Few-Weight Linear Codes Show Chinese title

Title: 某些少重量线性码的子域码

Abstract: Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $\bar{C}_{f,g}^{(q)}$ of six different families of linear codes $\bar{C}_{f,g}$ are presented, respectively. The parameters and weight distribution of the subfield codes and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also studied. Some of the resultant $q$-ary codes $\bar{C}_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes $\bar{C}_{f,g}$ are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-dual code are obtained with $m \geq 2$. As an application, several infinite families of 2-designs and 3-designs are also constructed with three families of linear codes of this paper. Show Chinese abstract

Abstract: 子域码在有限域上的线性码最近引起了广泛关注，因为其中一些码是最佳的，并且在秘密共享、认证码和关联方案中有应用。 在本文中，分别介绍了六种不同线性码族$\bar{C}_{f,g}$的$q$-元子域码$\bar{C}_{f,g}^{(q)}$。 子域码及其删余码$\bar{C}_{f,g}^{(q)}$的参数和重量分布被明确确定。 这些码的对偶码的参数也进行了研究。 其中一些结果$q$-元码$\bar{C}_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ 以及它们的对偶码是最佳的，有些具有已知的最佳参数。 该论文中第一类和第二类线性码$\bar{C}_{f,g}$的参数和权重枚举也被确定，其中第一类是一个满足 Griesmer 界的最优两权线性码，这两类码的对偶码是几乎 MDS 码。 作为本文的一个副产品，得到了一类$[2^{4m-2},2m+1,2^{4m-3}]$四元 Hermitian 自正交码，其参数为$m \geq 2$。 作为应用，还利用本文的三类线性码构造了几类无限的 2 设计和 3 设计。