Let Fq\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{q}$$\end{document} be the finite field with q=pm\documentclass[12pt]{minimal}
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\begin{document}$$q=p^{m}$$\end{document} elements, where p is an odd prime and m is a positive integer. For a positive integer t, let D⊂Fqt\documentclass[12pt]{minimal}
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\begin{document}$$D\subset \mathbb {F}^{t}_{q}$$\end{document} and let Trm\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {Tr}}_{m}$$\end{document} be the trace function from Fq\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{q}$$\end{document} onto Fp\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{p}$$\end{document}. In this paper, let D={(x1,x2,…,xt)∈Fqt\{(0,0,…,0)}:Trm(x1+x2+⋯+xt)=0},\documentclass[12pt]{minimal}
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\begin{document}$$D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},$$\end{document} we define a p-ary linear code CD\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C}_{D}$$\end{document} by CD={c(a1,a2,…,at):(a1,a2,…,at)∈Fqt},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$\end{document}where c(a1,a2,…,at)=(Trm(a1x12+a2x22+⋯+atxt2))(x1,x2,…,xt)∈D.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$\end{document}We shall present the complete weight enumerators of the linear codes CD\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C}_{D}$$\end{document} and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016).
