Cooperative Repair Schemes of Reed–Solomon Codes With Three Erasures

Reed-Solomon (RS) codes are widely utilized in distributed storage systems, where multiple node failures occur occasionally. For the cooperative repair of $[n,k]$ RS codes over $\mathbb {F}_{q^{\ell } }$ with three erasures, existing schemes necessitate $ n {-} k \geq q^{\ell - 1} $ , $\ell \geq 4$ and typically require three rounds of information interaction during the collaboration phase. This letter presents two key advancements. First, we develop an enhanced three-round cooperative repair scheme by using linearized polynomials corresponding to meticulously designed subspaces. Compared with the existing schemes, the proposed scheme maintains the same repair bandwidth while reducing the constraint to $ n {-} k \geq q^{m} $ with an integer $m\in \left [{{\frac {3}{4}\ell, \ell -1}}\right]$ , $\ell \geq 4$ . Second, we refine the repair scheme to achieve one-round collaboration, albeit with a minor increase in the repair bandwidth.