A class of linear codes and their complete weight enumerators

Abstract

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.