A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic

Let $q$ be a power of an odd prime $p$.  We prove the MDS conjecture over $\mathbb F_q$ in every dimension $k$ satisfying$$
 2\leqslant k\leqslant B(p,q)
 \quad\text{or}\quad
 q+2-B(p,q)\leqslant k\leqslant q,
 \qquad
 B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor.
$$For fixed $p$, the first interval is linear in $q$; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions.

The main algebraic input is valid over an arbitrary field of characteristic $p>0$.  We prove a vanishing theorem for determinant relations and deduce full row rank for Chowdhury's matrices throughout a larger interval of arc sizes.  At the conjectural size $|G|=2k-3+n$, this proves Chowdhury's full-row-rank conjecture without its $q$-dependent restriction.  After specialization to $\mathbb F_q$, combining this rank theorem with the Ball–Lavrauw construction yields the stated MDS range.

As consequences, for every odd prime power $q$, every normal rational curve in $\mathrm{PG}(N,q)$ is complete for $2\leqslant N\leqslant q-2$, and every projective Reed–Solomon code of length $q+1$ and dimension $2\leqslant k\leqslant q-2$ has covering radius $q-k$.