Let $x_1, x_2, \ldots, x_{n-1}$ be nonnegative even integers. Let $A=(a_{ij})$ be a $4k\times 4k$ matrix, where $k$ is a positive odd integer, such that:
1) The matrix $A$ is symmetric.
2) All of the entries of $A$ consist of linear combinations of the form $c_1x_1+c_2x_2+\ldots+c_{n-1}x_{n-1}$, where $c_i=\pm 1$ for each $1 \leq i \leq n-1$.
3) The variables $x_1, \ldots, x_{n-1}$ are all nonnegative even integers.
4) All of the diagonal entries are equal to $4kr$, where $r$ is some nonnegative integer.
5) No two expressions in the upper triangular entries are the same; that is, all $\binom{4k}{2}$ of them are distinct.
6) All of the off-diagonal entries are either equal to $0$ or $-4k$; that is, these expressions in terms of the $x_i$'s either equal $0$ or $-4k$.
7) The sum of the entries in each row of $A$ equals $0$. That is, each row has precisely $r$ entries that are equal to $-4k$.
I would like to show that the nullity of $A$ is equal to $1$, $2$, or $4k$, but I do not know if I have enough information. I want to get such a matrix in reduced row echelon form. Does anyone have any ideas?