Conjecture on the 2-ranks of submatrices of finite quadratic-space incidence matrices

Let $PG(n,q)$ be the $n$-dimensional projective space over $\mathbb{F}_{q}$, and let $\mathbf{A}$ be the incidence matrix arising from the quadratic form with parameter $\alpha$, where $\alpha$ is a nonzero square or non-square element of $\mathbb{F}_{q}$. Consider the four submatrices $\mathbf{A}_{11},\mathbf{A}_{12},\mathbf{A}_{21},\mathbf{A}_{22}$ from the first partition of $\mathbf{A}$, with $\mathbf{A}_{11}$ and $\mathbf{A}_{22}$ the diagonal blocks and $\mathbf{A}_{12},\mathbf{A}_{21}$ the off-diagonal blocks.

The 2-rank conjecture. The $2$-ranks satisfy the following:

• If $n$ is odd, $\mathbf{A}_{11}$ is full rank; if $n$ is even, the $2$-rank of $\mathbf{A}_{11}$ is one less than its order.
• $\mathbf{A}_{22}$ is always full rank.
• If $n\geq 3$ is odd, the $2$-rank of $\mathbf{A}_{12}$ and $\mathbf{A}_{21}$ is

$$q^{n-1}+q^{n-3}$$

when $\alpha$ is a nonzero square element, and

$$q^{n-1}+q^{n-3}+\cdots+q^{2}$$

when $\alpha$ is a non-square element. If $n$ is even, their $2$-rank is

$$q^{n-1}+q^{n-3}+\cdots+q.$$

The conjecture proposes uniform rank formulas for the incidence submatrices associated with finite quadratic spaces; the preceding discussion records several cases for $n=2$, while the unresolved cases and the general assertions are the subject of the paper.