Constructions for several classes of few-weight linear codes and their applications

In this paper, for any odd prime \begin{document}$ p $\end{document} and an integer \begin{document}$ m\ge 3 $\end{document}, several classes of linear codes with \begin{document}$ t $\end{document}-weight \begin{document}$ (t = 3,5,7) $\end{document} are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for \begin{document}$ m = 3 $\end{document}, a class of MDS codes with parameters \begin{document}$ [p,3,p-2] $\end{document} are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and \begin{document}$ s $\end{document}-sum sets for any odd \begin{document}$ s>1 $\end{document}.