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 ($t \in \{2, 3\}$).

翻译：近来，有限域上线性码的子域码受到了广泛关注。其中一些码是最优的，并在秘密共享、认证码和结合方案中具有应用。本文分别考虑并研究了有限域$F_{q^m}$上由两个函数$f, g$参数化的六类不同线性码族$C_{f,g}$的$q$元子域码$C_{f,g}^{(q)}$。明确确定了$C_{f,g}^{(q)}$及其缩短码$\bar{C}_{f,g}^{(q)}$的参数和（汉明）重量分布。还分析了这些码的对偶码的参数。一些由此得到的$q$元码$C_{f,g}^{(q)}$、$\bar{C}_{f,g}^{(q)}$及其对偶码是最优的，另一些则具有已知最佳参数。前两类线性码$C_{f,g}$的参数和重量计数函数也已确定，其中第一类码是满足Griesmer界的最优二重量线性码，这两类码的对偶码是几乎MDS码。作为本文的副产品，得到了一族$[2^{4m-2},2m+1,2^{4m-3}]$的四元Hermitian自对偶码，其中$m \geq 2$。在应用方面，我们证明了三类导出线性码可以产生若干无限族$t$设计（$t \in \{2, 3\}$）。