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Subfield codes of linear codes over finite fields have recently received muchattention. Some of these codes are optimal and have applications in secretesharing, authentication codes and association schemes. In this paper, the$q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linearcodes $C_{f,g}$ 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 ofthese 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 andsome have the best known parameters. The parameters and weight enumerators ofthe first two families of linear codes $C_{f,g}$ are also settled, among whichthe first family is an optimal two-weight linear code meeting the Griesmerbound, and the dual codes of these two families are almost MDS codes. As abyproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternaryHermitian self-dual code are obtained with $m \geq 2$. As an application, weshow that three families of the derived linear codes give rise to severalinfinite families of $t$-designs ($t \in \{2, 3\}$).

Subfield Codes of Several Few-Weight Linear Codes Parametrized by Functions and Their Consequences