The dimension formulas for the codes C(1,q) and C(2,q) over odd prime powers

Let $q$ be an odd prime power, and let $C(1,q)$ and $C(2,q)$ be the codes defined from the incidence structures in the paper. Their dimensions are the integers $\dim C(1,q)$ and $\dim C(2,q)$. Dimension conjecture.

$$\dim C(1,q)=\frac{1}{2}q^3-q^2+\frac{3}{2}q-1,$$ $$\dim C(2,q)=\frac{1}{2}q^3-\frac{5}{2}q^2+\frac{9}{2}q-\frac{7}{2}.$$

These formulas were obtained by interpolation from computed tables and are verified in the source for every odd prime power $q$ with $5\leq q\leq31$; their validity beyond that range is not established.