A class of linear codes and their complete weight enumerators

Let \begin{document}$ {\mathbb 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 $\end{document} is a positive integer. Let \begin{document}$ \operatorname{Tr}_m $\end{document} denote the trace function from \begin{document}$ {\mathbb F}_q $\end{document} onto \begin{document}$ {\mathbb F}_p $\end{document} , and the defining set \begin{document}$ D\subset {\mathbb F}_q^t $\end{document} , where \begin{document}$ t $\end{document} is a positive integer. In this paper, the set \begin{document}$ D = \{(x_1, x_2, \cdots, x_t)\in {\mathbb F}_q^t:\operatorname{Tr}_m(x_1^2+x_2^2+\cdots+x_t^2) = 0, \operatorname{Tr}_m(x_1+x_2+\cdots+x_t) = 1\} $\end{document} . Define the \begin{document}$ p $\end{document} -ary linear code \begin{document}$ {\mathcal C}_D $\end{document} by \begin{document}$ \begin{eqnarray*} {\mathcal C}_D = \{\textbf{c}(a_1, a_2, \cdots, a_t): (a_1, a_2, \cdots, a_t)\in {\mathbb F}_q^t\}, \end{eqnarray*} $\end{document} where \begin{document}$ \textbf{c}(a_1, a_2, \cdots, a_t) = (\operatorname{Tr}_m(a_1x_1+a_1x_2\cdots+a_tx_t))_{(x_1, \cdots, x_t)\in D}. $\end{document} We evaluate the complete weight enumerator of the linear codes \begin{document}$ {\mathcal C}_D $\end{document} , and present its weight distributions. Some examples are given to illustrate the results.