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 proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic $p>0$. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at $|G|=2k-3+n$ it proves Chowdhury's full-row-rank conjecture without the $q$-dependent restriction. Specialization to $\mathbb F_q$, together with the Ball--Lavrauw construction, gives the stated MDS range. We also prove that, 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 $q-k$.

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