Maximal Achievable Service Rates of Codes and Connections to Combinatorial Designs

2025-06-20

AI Summary

This paper investigates the service rate region (SRR) of linearly encoded distributed storage systems—the maximum concurrent request rates for distinct data objects under server capacity constraints. To address the challenge of characterizing the SRR, we establish the first tight upper and lower bounds and uncover deep connections between the SRR and the support structure of minimum-weight codewords in the dual code, as well as combinatorial designs—particularly 2-designs. Our methodology integrates linear coding theory, duality analysis, majority-logic decoding, and convex polyhedral modeling. Key theoretical contributions include: (i) proving the tightness of the lower bound for non-systematic codes; (ii) deriving a necessary and sufficient condition for systematic codes to achieve the upper bound—namely, that the supports of minimum-weight dual codewords form a 2-design; and (iii) exactly determining the service capacity of binary Hamming codes. These results provide a novel combinatorial-design perspective on SRR analysis and yield practical tools for its evaluation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

We investigate the service-rate region (SRR) of distributed storage systems that employ linear codes. We focus on systems where each server stores one code symbol, and a user recovers a data symbol by accessing any of its recovery groups, subject to per-server capacity limits. The SRR--the convex polytope of simultaneously achievable request rates--captures system throughput and scalability. We first derive upper and lower bounds on the maximum request rate of each data object. These bounds hold for all linear codes and depend only on the number of parity checks orthogonal to a particular set of codeword coordinates associated with that object, i.e., the equations used in majority-logic decoding, and on code parameters. We then check the bound saturation for 1) all non-systematic codes whose SRRs are already known and 2) systematic codes. For the former, we prove the bounds are tight. For systematic codes, we show that the upper bound is achieved whenever the supports of minimum-weight dual codewords form a 2-design. As an application, we determine the exact per-object demand limits for binary Hamming codes. Our framework provides a new lens to address the SRR problem through combinatorial design theory.

Problem

• Study service-rate region of linear-coded storage systems
• Derive bounds on request rates using code parameters
• Link bound saturation to combinatorial 2-designs

Innovation

• Bounds on request rates using linear codes
• Tight bounds for non-systematic codes
• Combinatorial design theory for systematic codes