Maximum-distance-separable (MDS) codes are widely used in distributed storage, yet naive repair of a single erasure in an $[n,k]$ MDS code downloads the entire contents of $k$ nodes. Minimum Storage Regenerating (MSR) codes (Dimakis et al., 2010) minimize repair bandwidth by contacting $d>k$ helpers and downloading only a fraction of data from each. Guruswami and Wootters first proposed a linear repair scheme for Reed-Solomon (RS) codes, showing that they can be repaired with lower bandwidth than the naive approach. The existence of RS codes achieving the MSR point (RS-MSR codes) nevertheless remained open until the breakthrough construction of Tamo, Barg, and Ye, which yields RS-MSR codes with subpacketization $\ell = s \prod_{i=1}^n p_i$, where $p_i$ are distinct primes satisfying $p_i \equiv 1 \pmod{s}$ and $s=d+1-k$. In this paper, we present an improved construction of RS-MSR codes by eliminating the congruence condition $p_i \equiv 1 \pmod{s}$. Consequently, our construction reduces the subpacketization by a multiplicative factor of $φ(s)^n$ ( $φ(\cdot)$ is Euler's totient function) and broadens the range of feasible parameters for RS-MSR codes.

翻译：最大距离可分（MDS）码被广泛应用于分布式存储中，然而，在$[n,k]$ MDS码中，对单个擦除进行朴素修复需要下载$k$个节点的全部内容。最小存储再生（MSR）码（Dimakis等人，2010）通过联系$d>k$个辅助节点，并且仅从每个节点下载部分数据，从而最小化修复带宽。Guruswami和Wootters首次提出了Reed-Solomon（RS）码的线性修复方案，表明它们可以用比朴素方法更低的带宽进行修复。然而，达到MSR点（RS-MSR码）的RS码的存在性问题一直悬而未决，直到Tamo、Barg和Ye的突破性构造出现，该构造产生的RS-MSR码具有子分组化$\ell = s \prod_{i=1}^n p_i$，其中$p_i$是满足$p_i \equiv 1 \pmod{s}$的不同素数，且$s=d+1-k$。在本文中，我们通过消除同余条件$p_i \equiv 1 \pmod{s}$，提出了一种改进的RS-MSR码构造。因此，我们的构造将子分组化降低了$φ(s)^n$的乘法因子（$φ(\cdot)$是欧拉函数），并拓宽了RS-MSR码的可行参数范围。