首页|A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic
A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic
Xiang Fan
A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic
A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic
摘要
Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over$\mathbb F_q$ in dimension $k$ whenever\[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q,\]where $B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor$.For fixed $p$, this gives a range of dimensions linear in $q$, in contrastwith the square-root scale of the previously known general unconditionalranges over proper extension fields, and proves the conjecture in anasymptotic proportion $1-1/(2p-3)$ of all dimensions.The main structural ingredient is a full-support obstruction for determinantrelations on arcs over arbitrary fields of odd characteristic. Afterprojection from $p-1$ points, local matching constraints followed bymultilinear descent force the resulting homogeneous system to have zerokernel. For a hypothetical $q+2$-point arc over $\mathbb F_q$, theBall--Lavrauw construction produces precisely the forbidden relations.The same framework also proves Chowdhury's conjecture that a family ofinclusion matrices has full row rank.