= Solution
Under $V=L$, the structure $L_{\omega_1}$ contains a <well-order code> for every <countable ordinal> and the representation of every well-order code that it contains. Given any $\alpha<\omega_1$, apply the <Downward Lowenheim-Skolem theorem> to choose a countable elementary substructure
$$
X\prec L_{\omega_1}
$$
containing $\alpha$. The <Mostowski collapse theorem> and condensation identify the transitive collapse of $X$ with $L_\lambda$, where $\lambda=X\cap\omega_1$ is a <limit ordinal> greater than $\alpha$.
Elementarity now verifies both coding properties. If a well-order code belongs to $L_\lambda$, its representation is carried into $L_\lambda$ by the collapse. Conversely, every $\beta<\lambda$ belongs to $X$, and elementarity supplies in $X$ a well-order code for $\beta$; the collapse fixes this code because it is a relation on $\omega$. Hence $L_\lambda$ is a <coding level of the constructible hierarchy>. Such $\lambda$ occur unboundedly below $\omega_1$, so $\Gamma$ has at least $\aleph_1$ elements; since $\Gamma\subseteq\omega_1$, it has exactly
$$
|\Gamma|=\aleph_1.
$$
Solved by gpt-5.6-sol high.
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