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：设$\mathbb{F}_q$上的$(n,k,\ell)$ MDS阵列码，在线性精确修复条件下，修复带宽和修复I/O能有多小？我们在域大小$q$、冗余度$r=n-k$和子分组化水平$\ell$固定而码长$n$变化的框架下研究此问题，并为此场景发展了几何方法。我们的出发点是将MDS阵列码的线性精确修复内在重构为子空间交集问题，对于修复I/O，则重构为由奇偶校验实现诱导的射影点配置。该视角通过简单的射影计数论证，建立了对$\mathbb{F}_q$上任意具有冗余度$r=n-k\ge 2$的$(n,k,\ell)$ MDS阵列码的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$r\ge 2$和子分组化水平$\ell$的此类下界。初看之下，射影计数界似乎相当粗糙，因此不太可能达到。我们证明当$r\ge 3$且$\ell\ge 2$时这一直觉正确。当$r=2$时，情况完全改变。利用有限几何中的德萨格展形，我们构造了在宽泛码长区间（直至最大可能长度$q^{\ell}+1$）上同时达到修复带宽和修复I/O下界的MDS阵列码。在最小非平凡情形$(r,\ell)=(2,2)$中，我们还证明了正则展形模型下的逆命题。这些结果共同揭示了支配线性精确修复的一致性障碍，并表明在双奇偶校验情形下该障碍是紧致的。