A Partial-Exclusion Repair Scheme for MDS Codes

2026-03-12

AI Summary

This work addresses the prohibitive sub-packetization complexity of traditional MDS codes, which require super-exponential sub-packetization to achieve the cut-set bound in single-node repair, severely hindering practical deployment. To overcome this limitation, we propose a Partial Exclusion (PE) repair framework that introduces tunable exclusion sets and enhanced node flexibility—breaking the conventional constraint of flexibility being fixed at one. Within this framework and leveraging scalar linear MDS code theory together with the algebraic structure of Reed–Solomon codes, we construct two classes of PE-RS codes. These codes attain the cut-set bound while achieving sub-packetization levels strictly below the existing lower bounds under high flexibility. The efficiency and practicality of the proposed repair process are further validated through Magma-based simulations.

Technology Category

• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

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.

Problem

• MDS codes
• sub-packetization
• repair scheme
• cut-set bound
• single-node repair

Innovation

• Partial-Exclusion repair
• sub-packetization
• MDS codes
• cut-set bound
• Reed-Solomon codes