  This paper investigates the existence, enumeration and asymptotic performance
of self-dual and LCD double circulant codes over Galois rings of characteristic
$p^2$ and order $p^4$ with $p$ and odd prime. When $p \equiv 3 \pmod{4},$ we
give an algorithm to construct a duality preserving bijective Gray map from
such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Using random coding, we obtain
families of asymptotically good self-dual and LCD codes over
$\mathbb{Z}_{p^2},$ for the metric induced by the standard
$\mathbb{F}_p$-valued Gray maps.
