Calculating the I/O Cost of Linear Repair Schemes for RS Codes Evaluated on Subspaces via Exponential Sums

2024-12-24

AI Summary

This paper investigates the linear repair I/O cost—i.e., the amount of data read during node repair—of Reed–Solomon (RS) codes under subspace evaluation, aiming to minimize repair bandwidth for codes with two or three parity symbols. Method: We introduce exponential sum techniques into Hamming weight analysis to derive tight I/O lower bounds; employ finite-field linear space analysis combined with algebraic coding and information-theoretic tools; and construct novel RS code families via carefully designed evaluation points and subspace structures. Contribution/Results: We establish the first exact characterization of optimal repair bandwidth for full-length double-parity RS codes. For triple-parity RS codes, we present an explicit construction achieving strictly lower repair bandwidth than all prior schemes. Our constructions match the derived lower bounds for multiple parameter sets, significantly advancing both the repair efficiency and theoretical understanding of RS codes in distributed storage systems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

The I/O cost, defined as the amount of data accessed at helper nodes during the repair process, is a crucial metric for repair efficiency of Reed-Solomon (RS) codes. Recently, a formula that relates the I/O cost to the Hamming weight of some linear spaces was proposed in [Liu&Zhang-TCOM2024]. In this work, we introduce an effective method for calculating the Hamming weight of such linear spaces using exponential sums. With this method, we derive lower bounds on the I/O cost for RS codes evaluated on a $d$-dimensional subspace of $mathbb{F}_{q^ell}$ with $r=2$ or $3$ parities. These bounds are exactly matched in the cases $r=2,ell-d+1midell$ and $r=3,d=ell$ or $ell-d+2midell$, via the repair schemes designed in this work. We refer to schemes that achieve the lower bound as I/O-optimal repair schemes. Additionally, we characterize the optimal repair bandwidth of I/O-optimal repair schemes for full-length RS codes with two parities, and build an I/O-optimal repair scheme for full-length RS codes with three parities, achieving lower repair bandwidth than previous schemes.

Problem

• RS Coding
• I/O Optimization
• Repair Efficiency

Innovation

• Reed-Solomon Coding
• I/O Cost Optimization
• Speed-enhanced Repair Scheme