Reed-Solomon code subpacketization limits for optimal linear repair found

In this paper, we study the minimum subpacketization of $(n,k)$ Reed-Solomon (RS) codes over $\mathbb{F}_{q^\ell}$ admitting cut-set-optimal exact $\mathbb{F}_q$-linear repair of every coordinate from any $d$ helpers. For $3\le k<d\le n-1$, write $s=d+1-k$ and denote this minimum by $\ell_{min}(n,k,d;q)$. Using divisibility and relative-degree constraints on cross-ratio fields of evaluation sets, we establish $\ell_{min}(n,k,d;q)\ge sG_s(d+1)$, where $G_s(m)$ is the minimum product of $m$ pairwise coprime integers, each at least $s$. Let $Δ_s(n,d)$ be the minimum of $lcm(a_1,\ldots,a_n)$ over positive integers $a_1,\ldots,a_n$ satisfying $lcm(a_j:j\in J\cup\{i\})\ge s~lcm(a_j:j\in J)$ for every $d$-subset $J\subseteq[n]$ and every $i\in[n]\setminus J$. We construct RS codes with subpacketization $sΔ_s(n,d)$ and further prove that $\ell_{min}(n,k,d;q)=sΔ_s(n,d)=sG_s(n)$ whenever $d(d+2)\geq 4(n+1)$, for every prime power $q$. In particular, this determines the minimum subpacketization in the full-helper case $d=n-1$, resolving the corresponding exact linear-repair case of Open Problem 20 posed by Ramkumar et al. in their survey on codes for distributed storage. We further give a finite formula for $G_s(n)$ and determine the exact threshold for its closed-form expression in terms of the first $n$ primes. For fixed $k$ and $d$, we prove $\log\ell_{min}(n,k,d;q)= Θ_{k,d}(\log n\log\log n)$ as $n\to\infty$, using divisor-function estimates and cover-free families. An affine-plane construction shows that $Δ_s(n,d)<G_s(n)$ for some parameters, so $sG_s(n)$ is not a universal formula. We conjecture that $\ell_{min}(n,k,d;q)=sΔ_s(n,d)$ for every prime power $q$ and all $3\le k<d\le n-1$.