The Subfield Codes of Some Few-Weight Linear Codes

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.