Repairing Reed-Solomon codes with low bandwidth is a central challenge in distributed storage. Following the trace-repair framework of Guruswami and Wootters (2017), recent works by Lin (2023) and Liu-Wan-Xing (2024) provided significant improvements in bandwidth using two distinct ideas. Lin constructed a trace-repair scheme that requires no contribution from a set of predetermined nodes $\mathscr{S}$, while Liu-Wan-Xing identified linear dependencies among the downloaded traces, relating the number of dependent traces to the dimension of a subspace $\mathscr{W}_k$. In this work, we fully utilize and unify these ideas. We compute the exact dimension of $\mathscr{W}_{k,\mathscr{S}}$ (a generalization of $\mathscr{W}_k$). We identify the trade-off between the set size $|\mathscr{S}|$ and the dimension $\dim(\mathscr{W}_{k,\mathscr{S}})$. We provide an algorithm to find the combination that results in the lowest bandwidth. Furthermore, we provide an explicit choice of the helper nodes for the repair. Finally, we prove that our optimized scheme never loses to the classical repair scheme, establishing a bandwidth guarantee of at most $k\log|\mathbb{F}|$ bits for all dimension $k$ and field $\mathbb{F}$, whenever the trace repair is applicable.

翻译：在分布式存储中，以低带宽修复里德-所罗门码是一个核心挑战。继Guruswami与Wootters（2017）的迹修复框架之后，Lin（2023）以及Liu-Wan-Xing（2024）的近作通过两种不同的思路，在带宽方面取得了显著改进。Lin构建了一种迹修复方案，其无需一组预定节点集$\mathscr{S}$的贡献；而Liu-Wan-Xing则识别了所下载迹之间的线性相关性，将相关迹的数量与子空间$\mathscr{W}_k$的维数联系起来。在本工作中，我们充分运用并统一了这些思想。我们计算了$\mathscr{W}_{k,\mathscr{S}}$（$\mathscr{W}_k$的推广）的精确维数。我们明确了集合大小$|\mathscr{S}|$与维数$\dim(\mathscr{W}_{k,\mathscr{S}})$之间的权衡关系。我们提供了一种算法来寻找能实现最低带宽的组合。此外，我们为修复过程给出了辅助节点的显式选择方案。最后，我们证明了我们优化后的方案绝不逊于经典修复方案，从而为所有维数$k$和域$\mathbb{F}$确立了至多$k\log|\mathbb{F}|$比特的带宽保证，只要迹修复适用。