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.

令q为奇质数，并用\（{\ mathbb {F}} _ q \）表示具有q个元素的有限域。在本文中，我们考虑环\（R = {\ mathbb {F}} _ q + u {\ mathbb {F}} _ q + v {\ mathbb {F}} _ q \），其中\（u ^ 2 = u，v ^ 2 = v，uv = vu = 0 \）并研究该环上的双循环和双负循环代码。我们首先获得双循环码为自对偶（分别为LCD）的必要和充分条件。然后，我们列举自对偶和LCD双循环，并在双negacirculant代码[R 。最后但并非最不重要的一点是，我们证明了R上的自对偶和LCD双循环码的灰度图像系列很好。以短长度给出了\（{\ mathbb {F}} _ 5 \）上的自对偶和LCD代码的几个数值示例，这些代码在R上的灰度图像。