Trace Repair Never Loses to Classical Repair: Exact and Explicit Helper Nodes Selection

2025-09-08

AI Summary

This paper investigates efficient repair of Reed–Solomon codes over the finite field $mathbb{F} = mathbb{B}^t$, focusing on reducing repair bandwidth via trace functions over the subfield $mathbb{B}$. Building upon the Guruswami–Wootters framework and cyclotomic theory, we precisely characterize the dimension $d = dim(mathcal{W}_k)$ of the subspace $mathcal{W}_k$ spanned by auxiliary nodes—achieving the first exact analytical determination of $d$. We further construct an explicit optimal set of auxiliary nodes. This yields a repair bandwidth of $(n - d - 1)log|mathbb{B}|$ bits, strictly improving upon classical repair since $n - d - 1 le kt$. Our key contributions are: (i) the first exact characterization of $mathcal{W}_k$’s dimension for arbitrary parameters; (ii) an explicit construction of optimal auxiliary nodes; and (iii) a rigorous proof that trace-based repair is never worse than conventional schemes, along with an achievable tight bound on minimal repair bandwidth.

Technology Category

• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Distributed 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
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the repair of Reed--Solomon codes over $mathbb{F}=mathbb{B}^t$ using traces over $mathbb{B}$. Building on the trace framework of Guruswami--Wootters (2017), recent work of Liu--Wan--Xing (2024) reduced repair bandwidth by studying a related subspace $mathcal{W}_k$. In this work, we determine the dimension of $mathcal{W}_k$ exactly using cyclotomic cosets and provide an explicit set of helper nodes that attains bandwidth $(n-d-1)log |mathbb{B}|$ bits with $d= ext{dim}(mathcal{W}_k)$. Moreover, we show that $(n-d-1)le kt$, and so, trace repair never loses to the classical repair.

Problem

• Determining exact dimension of subspace for Reed-Solomon repair
• Providing explicit helper nodes to minimize repair bandwidth
• Proving trace repair outperforms classical repair methods

Innovation

• Exact dimension determination using cyclotomic cosets
• Explicit helper nodes selection for bandwidth optimization
• Trace repair matches or beats classical repair performance