  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.
