Complete weight enumerators of two classes of linear codes

In this paper, we give the complete weight enumerators of two classes of linear codes over the finite field Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p}$\end{document}, where p is a prime. These linear codes are the torsion codes of MacDonald codes over the finite non-chain ring Fp+vFp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p}+v\mathbb {F}_{p}$\end{document}, where v2 = v. We also employ these linear codes to construct systematic authentication codes with new parameters.