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 $r=n-k$，且参数 $q$、$r$ 和 $\ell$ 固定，码长 $n$ 可变。近期提出的射影计数论证给出了该情形下修复带宽和修复I/O的一般下界。尽管此界在双校验情形下可在一宽泛码长区间内达到，但当 $r\ge 3$ 且 $\ell\ge 2$ 时无法达到。本文对该界背后的计数论证进行精化，建立了一个更紧的下界，我们称之为关联重数界。我们证明：对于 $\mathbb{F}_q$ 上任意 $(n,k,\ell)$ MDS阵列码（$r\ge 2$），其平均和最坏情况下的修复带宽及修复I/O均至少为 $$\ell(n-1)-(r-1)\frac{q^\ell-1}{q-1}.$$ 当 $r=2$ 时，此界与先前的射影计数界一致；当 $r\ge 3$ 时，则严格强于后者。我们还证明关联重数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 $(q-1)/(r-1)\ge 2$。则对于满足 $$2(r-1)\frac{q^\ell-1}{q-1}\le n\le q^\ell+1$$ 的任意整数 $n$，存在 $\mathbb{F}_q$ 上的 $(n,n-r,\ell)$ MDS阵列码，其修复带宽和修复I/O同时达到关联重数界。这些码通过正规有理曲线的域约化构造而来。综上，这些结果揭示了关联重数是超越双校验情形下MDS阵列码线性精确修复的主流几何原理。