  In this article, we study K3 double structures on minimal rational surfaces
$Y$. The results show there are infinitely many non-split abstract K3 double
structures on $Y = \mathbb{F}_e$ parametrized by $\mathbb P^1$, countably many
of which are projective. For $Y = \mathbb{P}^2$ there exist a unique non-split
abstract K3 double structure which is non-projective (see Dr\'ezet's article in
arXiv:2004.04921). We show that all projective K3 carpets can be smoothed to a
smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is
embedded as a variety of minimal degree, there are infinitely many embedded K3
carpet structures on $Y$. Moreover, we show any embedded projective K3 carpet
on $\mathbb F_e$ with $e<3$ arises as a flat limit of embeddings degenerating
to $2:1$ morphism. The rest do not, but we still prove the smoothing result. We
further show that the Hilbert points corresponding to the projective K3 carpets
supported on $\mathbb{F}_e$, embedded by a complete linear series are smooth
points if and only if $0\leq e\leq 2$. In contrast, Hilbert points
corresponding to projective K3 carpets supported on $\mathbb{P}^2$ and embedded
by a complete linear series are always smooth. The results in a recent paper of
Bangere, Gallego, and Gonz\'alez show that there are no higher dimensional
analogues of the results in this article.
