For an $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$, how small can the repair bandwidth and repair I/O be under linear exact repair? We study this question in the regime where the field size $q$, the redundancy $r=n-k$, and the sub-packetization level $\ell$ are fixed, while the code length $n$ varies, and we develop a geometric approach to this setting. Our starting point is an intrinsic reformulation of linear exact repair for MDS array codes in terms of subspace intersections and, for repair I/O, the projective point configurations induced by a parity-check realization. This viewpoint yields a simple projective counting argument establishing the general lower bound $$β_{\mathrm{avg}},β_{\max},γ_{\mathrm{avg}},γ_{\max}\;\ge\;\ell(n-1)-\frac{q^{(r-1)\ell}-1}{q-1}$$ for linear exact repair of every $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$ with redundancy $r=n-k\ge 2$. To our knowledge, this is the first lower bound of this form that applies to arbitrary redundancy $r\ge 2$ and sub-packetization level $\ell$. At first glance, the projective counting bound appears rather coarse and therefore unlikely to be attained. We prove that this intuition is correct whenever $r\ge 3$ and $\ell\ge 2$. For $r=2$, the picture changes completely. Using Desarguesian spreads from finite geometry, we construct MDS array codes that attain the bound over a broad interval of code lengths, up to the maximum possible length $q^{\ell}+1$, and do so simultaneously for both repair bandwidth and repair I/O. In the smallest nontrivial case $(r,\ell)=(2,2)$, we also prove a converse within the regular-spread model. Together, these results identify a uniform obstruction governing linear exact repair and show that, in the two-parity case, this obstruction is tight.

翻译：对于定义在 $\mathbb{F}_q$ 上的 $(n,k,\ell)$ MDS 阵列码，在线性精确修复中，修复带宽和修复 I/O 能达到多小？我们研究在场大小 $q$、冗余度 $r=n-k$ 和子分组层级 $\ell$ 固定而码长 $n$ 变化的情况下这一问题，并为此建立了一种几何方法。我们的出发点是将 MDS 阵列码的线性精确修复问题本质地重新表述为子空间交以及（对于修复 I/O）由奇偶校验实现诱导的射影点构型。这一视角通过简单的射影计数论证，给出了关于 $\mathbb{F}_q$ 上任意冗余度 $r=n-k\ge 2$ 的 $(n,k,\ell)$ MDS 阵列码线性精确修复的通用下界：$$β_{\mathrm{avg}},β_{\max},γ_{\mathrm{avg}},γ_{\max}\;\ge\;\ell(n-1)-\frac{q^{(r-1)\ell}-1}{q-1}$$ 据我们所知，这是首个适用于任意冗余度 $r\ge 2$ 和子分组层级 $\ell$ 的此类下界。初看之下，该射影计数界显得相当粗略，因此似乎难以达到。我们证明，当 $r\ge 3$ 且 $\ell\ge 2$ 时，这一直觉是正确的。而对于 $r=2$，情况则完全不同。利用有限几何中的德萨格扩散，我们构造了在码长广泛区间（直至最大可能长度 $q^{\ell}+1$）内同时达到该下界（针对修复带宽和修复 I/O）的 MDS 阵列码。在最小非平凡情形 $(r,\ell)=(2,2)$ 下，我们还证明了正则扩散模型内的逆命题。综合这些结果，我们识别出了控制线性精确修复的一个统一障碍，并表明在双奇偶校验情形下该障碍是紧的。