Consider an $n \times n \times n \times\dots\times n$ torus board of total size $n^k$ with $n > 4$ either even or odd.

Consider the basic cube of size $1 \times 1 \times \dots \times 1$ at a lattice point on the board and label the vertices from $\Bbb F_2$ with the vertices of the cube taking value $1$ and rest of the board taking value $0$. Translate the cube over each of the lattice points to get $n^k$ distinct $F_2$ configurations of the board. Call the set of configurations as $C$.

One can take $2^{n^k}$ possible $F_2$ linear combinations of the members of $C$ of which many resulting configurations would be the same. Call the new set of configurations $D$.

My questions:

1. Is there a recurrence formula for the number of distinct elements of $D$ in terms of $n$ and $k$? What is $|D|$, the cardinality of $D$?

2. If the number of elements in any linear combination is restricted to be exactly $t \leq n^k$, then call the resulting configuration $D_{t}$. Is there a recurrence formula for the number of distinct elements of $D_{t}$ in terms of $t$, $n$ and $k$? What is $|D_t|$, the cardinality of $D_t$?

3. Let $N_t$ be the maximum number of lattice points of any member of $D_t$. What is $\frac{N_t}{t}$ as a function of $t$, $n$ and $k$? (Example: $\frac{N_2}{2} = 2^k$ for all $k$ and $n > 4$).

4. Let for a given $n$ and $k$, the maximum value of $\frac{N_t}{t}$ be at $N_t = N_{n,k}$ and $t = t_{n,k}$. What is $N_{n,k}$ and $t_{n,k}$?