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

Double circulant codes of length $2n$ over the semilocal ring $R = \mathbb{F}q + u\mathbb{F}q,\, u^2=u,$ are studied when $q$ is an odd prime power, and $-1$ is a square in $\mathbb{F}q.$ Double negacirculant codes of length $2n$ are studied over $R$ when $n$ is even and $q$ is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length $2n$ is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length $4n$ over $\mathbb{F}q$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below. The parameters of examples of the modest length are computed. Several such codes are optimal.