Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

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Let q be an odd prime power, and denote by $${\mathbb {F}}_q$$ the finite field with q elements. In this paper, we consider the ring $$R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$ , 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 $${\mathbb {F}}_5$$ as the Gray images of these codes over R are given in short lengths.