发表机构
School of Mathematics, Sun Yat-sen University（中山大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明奇特征有限域上MDS猜想在维度随q线性增长的区间内成立，渐近覆盖所有维度的1-1/(2p-3)比例，并证明Chowdhury包含矩阵满秩猜想。
AI 中文摘要
设 $q$ 为奇素数 $p$ 的幂。我们证明当 $2\leqslant k\leqslant B(p,q)$ 或 $q+2-B(p,q)\leqslant k\leqslant q$ 时，$\mathbb F_q$ 上维度 $k$ 的 MDS 猜想成立，其中 $B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor$。对于固定的 $p$，这给出了维度范围随 $q$ 线性增长的结果，与先前已知的关于真扩域的一般无条件范围的平方根尺度形成对比，并在所有维度中渐近比例为 $1-1/(2p-3)$ 的范围内证明了该猜想。主要的结构性成分是奇特征任意域上弧的行列式关系的全支撑障碍。在从 $p-1$ 个点投影后，局部匹配约束随后进行多重线性下降，迫使所得齐次系统具有零核。对于 $\mathbb F_q$ 上假设的 $q+2$ 点弧，Ball--Lavrauw 构造恰好产生了禁止的关系。同一框架还证明了 Chowdhury 的猜想，即包含矩阵族具有满行秩。
英文摘要
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 $q-k$.
Comments23 pages. The proof has been reorganized to make its homological structure explicit. Other versions of this preprint are available on Zenodo: https://doi.org/10.5281/zenodo.22908066
