A linear-in-q range of dimensions for the MDS conjecture over Fq in odd characteristic

Let q be a power of an odd prime p. We prove the MDS conjecture over Fq in every dimension k satisfying2 k B(p,q) q+2-B(p,q) k q, B(p,q)=(p-2)q+6p-10/2p-3.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 Fq, 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 PG(N,q) is complete for 2 N q-2, and every projective Reed–Solomon code of length $q+1$ and dimension 2 k q-2 has covering radius $q-k$.