An effective minimal encoding of uncountable sets

We propose a method for encoding sets of the countable ordinals by generic reals which preserves cardinality and enjoys the property of minimality over the encoded set.
For $W\subseteq\omega_1$ there is a cardinal-preserving generic extension $L[W][x]$ of the class $L[W]$ by a generic real $x$ such that $W$ belongs to the class $L[x]$, i.e., $W$ is Gödel constructible with respect to $x$, while $x$ itself is minimal over $L[W]$.