Title: Subfield Codes of Several Few-Weight Linear Codes Parametrized by Functions and Their Consequences Show Chinese title

Title: 由函数参数化的几个低权重线性码的子域码及其后果

Abstract: Subfield codes of linear codes over finite fields have recently received much attention. 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 $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. Some of the resultant $q$-ary codes $C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ 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 $C_{f,g}$ 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, we show that three families of the derived linear codes give rise to several infinite families of $t$-designs ($t \in \{2, 3\}$). Show Chinese abstract

Abstract: Subfield codes of linear codes over finite fields have recently received much attention. 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 $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. 一些结果的$q$-元码$C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ 及其对偶码是最佳的，有些具有最好的已知参数。线性码$C_{f,g}$的前两族的参数和重量枚举也被确定，其中第一族是一个满足 Griesmer 界的最佳二重量线性码，这两族的对偶码是几乎 MDS 码。作为本文的副产品，获得了一族$[2^{4m-2},2m+1,2^{4m-3}]$四元 Hermitian 自对偶码，其具有$m \geq 2$。 作为应用，我们证明了三个导出线性码族可以产生若干无限族的$t$-设计 ($t \in \{2, 3\}$)。