We study linear exact repair for $(n,k,\ell)$ MDS array codes over $\mathbb{F}_q$, with redundancy $r=n-k$, in the regime where $q$, $r$, and $\ell$ are fixed and the code length $n$ varies. A recent projective counting argument gives a general lower bound on repair bandwidth and repair I/O in this setting. While this bound is attained over a broad interval of code lengths in the two-parity case, it is not attained once $r\ge 3$ and $\ell\ge 2$. In this paper, we refine the counting argument behind this bound and establish a sharper lower bound, which we call the incidence-multiplicity bound. We prove that for every $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$ with $r\ge 2$, both the average and worst-case repair bandwidth, as well as the average and worst-case repair I/O, are at least $$\ell(n-1)-(r-1)\frac{q^\ell-1}{q-1}.$$This bound agrees with the earlier projective counting bound when $r=2$, and is strictly stronger for every $r\ge 3$. We also show that the incidence-multiplicity bound is sharp in a broad parameter range. Assume that $\ell\ge 2$, $r\ge 2$, $(r-1)\mid(q-1)$, and $(q-1)/(r-1)\ge 2$. Then for every integer $n$ satisfying $$2(r-1)\frac{q^\ell-1}{q-1}\le n\le q^\ell+1,$$ there exists an $(n,n-r,\ell)$ MDS array code over $\mathbb{F}_q$ that attains the incidence-multiplicity bound simultaneously for both repair bandwidth and repair I/O. These codes arise from field reduction of a normal rational curve. Together, these results reveal incidence multiplicity as the governing geometric principle for linear exact repair in MDS array codes beyond the two-parity case.

翻译：我们研究$\mathbb{F}_q$上$(n,k,\ell)$ MDS阵列码的线性精确修复问题，其中冗余度$r=n-k$，并考虑$q$、$r$和$\ell$固定而码长$n$变化的情形。近期提出的射影计数论证给出了该设定下修复带宽和修复I/O的通用下界。尽管在双奇偶校验情形下该下界可在较宽的码长区间内达到，但当$r\ge 3$且$\ell\ge 2$时却无法达到。本文改进了该下界背后的计数论证，建立了更紧的下界，我们称之为关联重数下界。我们证明：对于$\mathbb{F}_q$上任意满足$r\ge 2$的$(n,k,\ell)$ MDS阵列码，其平均修复带宽与最坏情形修复带宽，以及平均修复I/O与最坏情形修复I/O均至少为$$\ell(n-1)-(r-1)\frac{q^\ell-1}{q-1}.$$当$r=2$时该下界与原有射影计数下界一致，而对所有$r\ge 3$情形则严格更强。我们还证明关联重数下界在广泛参数范围内是紧的。假设$\ell\ge 2$，$r\ge 2$，$(r-1)\mid(q-1)$且$(q-1)/(r-1)\ge 2$，则对每个满足$$2(r-1)\frac{q^\ell-1}{q-1}\le n\le q^\ell+1$$的整数$n$，都存在$\mathbb{F}_q$上的$(n,n-r,\ell)$ MDS阵列码，其修复带宽与修复I/O同时达到关联重数下界。这些码由正规有理曲线的域约化构造得到。综合以上结果，揭示了关联重数是超越双奇偶校验情形下MDS阵列码线性精确修复的核心几何原理。