For scalar maximum distance separable (MDS) codes, the conventional repair schemes that achieve the cut-set bound with equality for the single-node repair have been proven to require a super-exponential sub-packetization level.As is well known, such an extremely high level severely limits the practical deployment of MDS codes.To address this challenge, we introduce a partial-exclusion (PE) repair scheme for scalar linear codes.In the proposed PE repair framework, each node is associated with an exclusion set.The cardinality of the exclusion set is called the flexibility of the node.The maximum value of flexibility over all nodes defines the \textit{flexibility} of the PE repair scheme. Notably, the conventional repair scheme is the special case of PE repair scheme where the flexibility is 1. Under the PE repair framework, for any valid flexibility, we establish a lower bound on the sub-packetization level of MDS codes that meet the cut-set bound with equality for single-node repair. To realize MDS codes attaining the cut-set bound under the PE repair framework, we propose two generic constructions of Reed-Solomon (RS) codes. Moreover, we demonstrate that for a sufficiently large flexibility, the sub-packetization level of our constructions is strictly lower than the known lower bound established for the conventional repair schemes.This implies that, from the perspective of sub-packetization level, our constructions outperform all existing and potential constructions designed for conventional repair schemes. Finally, we implement the repair process for these codes as executable Magma programs, thereby exhibiting the practical efficiency of our constructions.

翻译：对于标量最大距离可分（MDS）码，已知实现单节点修复割集界等号的传统修复方案已被证明需要超指数级的子分组化水平。众所周知，这种极高的水平严重限制了MDS码的实际部署。为应对这一挑战，我们针对标量线性码提出了一种部分排除（PE）修复方案。在所提出的PE修复框架中，每个节点关联一个排除集。排除集的基数称为节点的灵活性。所有节点灵活性的最大值定义为PE修复方案的\textit{灵活性}。值得注意的是，传统修复方案是灵活性为1的PE修复方案的特例。在PE修复框架下，对于任意有效灵活性，我们建立了满足单节点修复割集界等号的MDS码子分组化水平的下界。为实现PE修复框架下达到割集界的MDS码，我们提出了两种Reed-Solomon（RS）码的通用构造。此外，我们证明对于足够大的灵活性，我们构造的子分组化水平严格低于传统修复方案已知下界。这意味着，从子分组化水平的角度看，我们的构造优于所有为传统修复方案设计的现有及潜在构造。最后，我们将这些码的修复过程实现为可执行的Magma程序，从而展示了我们构造的实际效率。