On the Service Rate Region of Reed-Muller Codes

2025-01-22

AI Summary

This work characterizes the service rate region (SRR) of Reed–Muller (RM) codes in distributed storage systems, quantifying their capacity to serve concurrent user read requests while ensuring efficient data repair. We develop a unified analytical framework integrating algebraic coding theory and convex geometry. First, we establish foundational theory on the existence, uniqueness, and complete enumeration of recovery sets for RM codes. Second, we uncover a one-to-one correspondence between recovery sets and minimum-weight codewords in the dual RM code. Third, we derive a tight upper bound on the largest simplex contained within the SRR. As a key result, we obtain an explicit closed-form expression for the maximum achievable request rate for a single data object—significantly improving the precision of SRR characterization. This yields the first theoretical optimality guarantee for RM-coded storage systems under high-concurrency, low-latency access requirements.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We study the Service Rate Region (SRR) of Reed-Muller (RM) codes in the context of distributed storage systems. The SRR is a convex polytope comprising all achievable data access request rates under a given coding scheme. It represents a critical metric for evaluating system efficiency and scalability. Using the geometric properties of RM codes, we characterize recovery sets for data objects, including their existence, uniqueness, and enumeration. This analysis reveals a connection between recovery sets and minimum-weight codewords in the dual RM code, providing a framework for identifying small recovery sets. Using these results, we derive explicit and tight bounds for the maximal achievable demand for individual data objects, which define the maximal simplex within the service rate region.

Problem

• Reed-Muller Codes
• Storage Systems
• Data Repair

Innovation

• Reed-Muller Codes
• Service Rate Region
• Data Loss Recovery