📝 SummarXiv

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 $d=\text{dim}(\mathcal{W}_k)$. Moreover, we show that $(n-d-1)\le kt$, and so, trace repair never loses to the classical repair.