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

Abstract

Repairing Reed-Solomon codes with low bandwidth is a central challenge in distributed storage. Following the trace-repair framework of Guruswami and Wootters (2017), recent works by Lin (2023) and Liu-Wan-Xing (2024) provided significant improvements in bandwidth using two distinct ideas. Lin constructed a trace-repair scheme that requires no contribution from a set of predetermined nodes $\mathscr{S}$, while Liu-Wan-Xing identified linear dependencies among the downloaded traces, relating the number of dependent traces to the dimension of a subspace $\mathscr{W}_k$. In this work, we fully utilize and unify these ideas. We compute the exact dimension of $\mathscr{W}_{k,\mathscr{S}}$ (a generalization of $\mathscr{W}_k$). We identify the trade-off between the set size $|\mathscr{S}|$ and the dimension $\dim(\mathscr{W}_{k,\mathscr{S}})$. We provide an algorithm to find the combination that results in the lowest bandwidth. Furthermore, we provide an explicit choice of the helper nodes for the repair. Finally, we prove that our optimized scheme never loses to the classical repair scheme, establishing a bandwidth guarantee of at most $k\log|\mathbb{F}|$ bits for all dimension $k$ and field $\mathbb{F}$, whenever the trace repair is applicable.