Maximum distance separable (MDS) codes are widely used in distributed storage, but naively repairing a single failure in an $(n,k)$ MDS code requires downloading the full contents of $k$ surviving nodes. Minimum storage regenerating (MSR) codes, introduced by Dimakis et al., minimize repair bandwidth while preserving the MDS property by contacting $d>k$ helper nodes and downloading only a fraction of each helper. For scalar MDS codes, Guruswami and Wootters established a linear repair framework, and Tamo, Ye, and Barg subsequently gave the first explicit Reed-Solomon (RS) codes achieving the MSR point. Their construction yields RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$, where $s=d+1-k$ and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{s}$. In this paper, we show that this congruence condition is not intrinsic to the RS repair problem. We develop a basis-transformation approach to the construction of repair-enabling subspaces. The approach consists of three deterministic operations -- Euclidean Square Partition, Transposition, and Column Aggregation -- which construct the required repair-enabling subspaces directly from the standard monomial basis of the repair field. Consequently, we obtain RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$ for arbitrary distinct primes $p_i>s$. For fixed $s$, this improves the subpacketization of the Tamo--Ye--Barg construction by a factor asymptotic to $\varphi(s)^{n+\mathrm{o}(n)}$, where $\varphi(\cdot)$ denotes Euler's totient function.

翻译：最大距离可分（MDS）码广泛应用于分布式存储，但直接修复$(n,k)$ MDS码中的单个故障需要下载$k$个存活节点的全部内容。Dimakis等人提出的最小存储再生（MSR）码通过连接$d>k$个辅助节点并仅下载每个辅助节点的一部分数据，在保持MDS性质的同时最小化修复带宽。对于标量MDS码，Guruswami和Wootters建立了线性修复框架，随后Tamo、Ye和Barg给出了首个达到MSR点的显式Reed-Solomon（RS）码。他们的构造产生了子分组长度$\ell=s\prod_{i=1}^n p_i$的RS-MSR码，其中$s=d+1-k$，不同素数$p_i$满足$p_i\equiv 1\pmod{s}$。本文证明该同余条件并非RS修复问题的本质属性。我们提出一种基于基变换的修复使能子空间构造方法，该方法包含三个确定性操作——欧几里得平方划分、转置和列聚合——直接从修复域的单项式标准基构造所需的修复使能子空间。由此，对于任意满足$p_i>s$的不同素数$p_i$，我们获得子分组长度$\ell=s\prod_{i=1}^n p_i$的RS-MSR码。对于固定$s$，该构造将Tamo-Ye-Barg构造的子分组长度改进了渐近因子$\varphi(s)^{n+\mathrm{o}(n)}$，其中$\varphi(\cdot)$表示欧拉函数。