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.

翻译：我们研究了在低通信带宽条件下，Reed-Solomon码中单个擦除的鲁棒修复问题。聚焦于Guruswami-Wootters迹修复框架，我们探究了当辅助节点返回错误响应时，能否正确修复失效节点。等价地，我们将下载的迹集合视为一种码，称之为修复迹码。通过利用分圆陪集刻画关联多项式的零系数，我们推导出允许纠正给定数量错误迹$e$的维度$k$的上界，以及作为$k$函数的最小距离下界。对于$q=2$的情况，我们利用分圆陪集代表元的显式公式，得到了单错误纠正的最优维度界。我们还提出了两种高效的鲁棒修复方案。第一种方案达到了BCH界保证的纠错能力。为逼近基于特征和表示的更强界，我们开发了第二种方案，该方案能以计算复杂度增加一个因子$n$为代价容忍更多错误。