Complete weight enumerator of torsion codes

<p style='text-indent:20px;'>In this paper, we introduce two classes of MacDonald codes over the finite non-chain ring <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p $\end{document}</tex-math></inline-formula> and their torsion which are linear id="M2">\begin{document}$ \mathbb{F}_p $\end{document}</tex-math></inline-formula>, where id="M3">\begin{document}$ p is an odd prime id="M4">\begin{document}$ v^3 = v $\end{document}</tex-math></inline-formula>. We give complete weight enumerator codes. As application, systematic authentication obtained by these codes.</p>