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Let q be an odd prime power, and denote by F q the finite field with q elements. In this paper, we consider the ring R = F q + uF q + vF q , where u 2 = u, v 2 = v, uv = vu = 0 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 F 5 as the Gray images of these codes over R are given in short lengths.

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}_4 $\end{document}</tex-math></inline-formula> be the ring of integers modulo <inline-formula><tex-math id="M3">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>. This paper studies mixed alphabets <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}_4\mathbb{Z}_4[u^3] $\end{document}</tex-math></inline-formula>-additive cyclic and <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic codes for units <inline-formula><tex-math id="M6">\begin{document}$ \lambda = 1+2u^2,3+2u^2 $\end{document}</tex-math></inline-formula>. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z}_4 $\end{document}</tex-math></inline-formula>-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.</p>

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