Self-dual double circulant, self-dual double negacirculant and LCD double negacirculant codes over the ring F[]/⟨2-, 2-, ⟩

In this paper, we investigate self-dual double circulant, and self-dual and linear complementary dual (LCD) double negacirculant codes over a finite ring R = F_q + u F_q + v F_q + uv F_q , where u^2=u , v^2=v , uv=vu and q=p^m. We study the algebraic structure of double circulant codes over R. We provide necessary and sufficient conditions for a double circulant code to be a self-dual code. We give a formula to get the total number of self-dual double circulant codes over the ring R. We compute distance bounds for self-dual double circulant codes over R. In addition, by using a Gray map, we show that the families of self-dual double circulant codes under the Gray map are asymptotically good. Moreover, the algebraic structure of double negacirculant codes and necessary and sufficient conditions for a double negacirculant code to be a self-dual code and to be an LCD code are also given. We determine the total number of self-dual and LCD double negacirculant codes over R.