Given a graph $G = (V, E)$, a distance $d$-independent set is a subset $S \subseteq V$ such that any two vertices $x, y \in S$ have distance at least $d$. Thus traditional independent sets are actually distance $2$-independent sets. Denote by $\alpha_d(G)$ the maximum size of such a set in $G$.
I'm interested in $\alpha_d(Q_n)$, where $Q_n$ is the hypercube of dimension $n$ (i.e. $V(Q_n)$ is the set of all length $n$ binary strings, and two strings share an edge iff they differ by one bit). We know about $\alpha_2(Q_n)$. Is something known for larger $d$.
If not, is anything known specifically for $d = n/2$ ? This is like asking how many strings can I find such that they are all have Hamming distance at least $n/2$.