Given $d\in\Bbb N$, pick $N=2^{2d}$ distinct $a_j$ from $\big\{1,\dots,2^{d^2}-1\big\}$ and pick $i$ from $\big\{3,\dots,2^{d}\big\}$.
On average how many of $i$-subsets in $\big\{\alpha^{a_j}\big\}_{j=1}^{N}$ sum to $0$ where $\alpha$ generates $\Bbb F_{2^{d^2}}$?