Some two-weight and three-weight linear codes

Let \begin{document}$\Bbb F_q$\end{document} be the finite field with \begin{document}$q = p^m$\end{document} elements, where \begin{document}$p$\end{document} is an odd prime and \begin{document}$m$\end{document} is a positive integer. For a positive integer \begin{document}$t$\end{document} , let \begin{document}$D \subset \Bbb F_q^t$\end{document} and let \begin{document}$\mbox{Tr}_m$\end{document} be the trace function from \begin{document}$\Bbb F_q$\end{document} onto \begin{document}$\Bbb F_p$\end{document} . We define a \begin{document}$p$\end{document} -ary linear code \begin{document}$\mathcal C_D$\end{document} by \begin{document} $ \mathcal C_D = \{\textbf{c}(a_1,a_2, ..., a_t): a_1, a_2, ..., a_t ∈ \Bbb F_{p^m}\}, $ \end{document} where \begin{document}$\textbf{c}(a_1,a_2, ..., a_t) = \big(\mbox{Tr}_m(a_1x_1+a_2x_2+···+a_tx_t)\big)_{(x_1,x_2, ..., x_t)∈ D}.$ \end{document} In this paper, we will present the weight enumerators of the linear codes \begin{document}$\mathcal C_D$\end{document} in the following two cases: 1. \begin{document}$D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t \setminus \{(0,0, ..., 0)\}: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 0\}$\end{document} ; 2. \begin{document}$D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 1\}$\end{document} . It is shown that \begin{document}$\mathcal C_D$\end{document} is a two-weight code if \begin{document}$tm$\end{document} is even and three-weight code if \begin{document}$tm$\end{document} is odd in both cases. The weight enumerators of \begin{document}$\mathcal C_D$\end{document} in the first case generalize the results in [ 17 ] and [ 18 ]. The complete weight enumerators of \begin{document}$\mathcal C_D$\end{document} are also investigated.