Linear codes over F2×(F2+vF2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)$$\end{document} and the MacWilliams identities

In this work, we study linear codes over the ring F2×(F2+vF2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)$$\end{document} and their weight enumerators, where v2=v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^2=v$$\end{document}. We first give the structure of the ring and investigate linear codes over this ring. We also define two weights called Lee weight and Gray weight for these codes. Then we introduce two Gray maps from F2×(F2+vF2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)$$\end{document} to F23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_2^3$$\end{document} and study the Gray images of linear codes over the ring. Moreover, we prove MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators.