Droms–Mellinger–Meyer dimension conjecture for projective-plane LDPC codes

Let $q$ be an odd prime power, and let ${\mathbf B}$ and ${\mathbf B}_0$ be the incidence matrices of internal points versus secant lines and external points versus passant lines, respectively, in ${\mathrm{PG}}(2,q)$. Let $\mathcal{L}$ and $\mathcal{L}_0$ be their column $\mathbb F_2$-null spaces. Droms–Mellinger–Meyer's conjecture. The dimensions are

$$\dim_{\mathbb F_2}(\mathcal{L})=\begin{cases}\frac{q^2-1}{4}-q,&q\equiv1\pmod4,\frac{q^2-1}{4}-q+1,&q\equiv3\pmod4, \end{cases}$$

and

$$\dim_{\mathbb F_2}(\mathcal{L}_0)=\begin{cases}\frac{q^2-1}{4},&q\equiv1\pmod4,\frac{q^2-1}{4}+1,&q\equiv3\pmod4. \end{cases}$$

These formulas concern the dimensions of the four low-density parity-check codes constructed from submatrices of the point-line incidence matrix of ${\mathrm{PG}}(2,q)$. The article confirms this conjecture, so the claim is solved.