Self-dual and LCD double circulant and double negacirculant codes over Fq+uFq+vFq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$\end{document}

Let q be an odd prime power, and denote by Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q$$\end{document} the finite field with q elements. In this paper, we consider the ring R=Fq+uFq+vFq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$\end{document}, where u2=u,v2=v,uv=vu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^2=u, v^2=v,uv=vu=0$$\end{document} and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over F5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_5$$\end{document} as the Gray images of these codes over R are given in short lengths.