Robust Repair of Reed-Solomon Codes

2026-06-03

AI Summary

This work addresses the robust repair problem for Reed–Solomon codes under a single erasure in low-bandwidth settings, where helper nodes may return erroneous responses. Building upon the Guruswami–Wootters trace repair framework, the downloaded traces are interpreted as a “repair trace code,” and this paper establishes, for the first time, theoretical bounds on its dimension and minimum distance. By analyzing cyclotomic cosets and zero coefficients of associated polynomials, the authors derive an upper bound on the dimension and a lower bound on the minimum distance when tolerating up to $e$ erroneous traces. Notably, for $q=2$, they obtain the exact optimal dimension for single-error correction. Furthermore, two efficient repair schemes are proposed: one leveraging the BCH bound to achieve standard error-correction capability, and another exploiting character sum bounds to attain enhanced fault tolerance at the cost of an $n$-fold increase in computational complexity.

Abstract

We study the problem of robust repair of a single erasure in Reed--Solomon codes under low communication bandwidth. Focusing on the Guruswami--Wootters trace repair framework, we investigate whether a failed node can be correctly repaired in the presence of erroneous responses from helper nodes. Equivalently, we view the collection of downloaded traces as a code, which we call the repair-trace code. By characterizing the zero coefficients of the associated polynomial in terms of cyclotomic cosets, we derive upper bounds on the dimension $k$ that allow correction of a given number of erroneous traces $e$, as well as lower bounds on the minimum distance as a function of $k$. For the case $q=2$, we exploit explicit formulas for cyclotomic coset representatives to obtain the exact optimal dimension bound for single-error correction. We also propose two efficient robust repair schemes. Our first scheme achieves the error-correction capability guaranteed by the BCH bound. To approach a stronger bound based on character sums, we develop a second scheme that tolerates more errors at the cost of an additional factor $n$ in computational complexity.

Problem

• Reed-Solomon codes
• robust repair
• erasure correction
• error tolerance
• repair-trace code

Innovation

• Reed-Solomon codes
• robust repair
• repair-trace code
• cyclotomic cosets
• error correction