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

We study the repair of Reed--Solomon codes over F=Bᵗ using traces over 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 Wₖ. In this work, we determine the dimension of Wₖ exactly using cyclotomic cosets and provide an explicit set of helper nodes that attains bandwidth (n-d-1) |B| bits with d=dim (Wₖ). Moreover, we show that (n-d-1) kt, and so, trace repair never loses to the classical repair.