In distributed storage, cooperative repair is to simultaneously recoverh(h> 1) node erasures by downloading data from surviving nodes as well as collaboration between thehreplacement nodes. In this work, we propose a generalized cooperative repair framework for Reed-Solomon (RS) codes with two erasures. The key idea is to construct parity-check polynomials for the two replacement nodes respectively and then reduce the repair problem to the design of a linearized permutation polynomial related to the parity-check polynomials. We provide constructions of the linearized permutation polynomial in several cases, leading to cooperative repair schemes accordingly. Compared with the schemes given by Dauet al. 2021, our schemes retain the same repair bandwidth while apply to a much wider parameter regime and need only one-round collaboration. Finally we further reduce the repair bandwidth by the lifting method for RS codes of short length.

Piggybacking design has been applied extensively in distributed storage systems in recent years, since it can reduce repair bandwidth significantly with small sub-packetization. In this work, a bidirectional piggybacking design (BPD) is proposed with sub-packetization$2\leq l\leq r$, where$r=n-k$equals the redundancy of an$[n,k]$linear code. Unlike most existing piggybacking designs, there is no distinction between systematic nodes and parity nodes in BPD and the piggybacks are added bidirectionally. Consequently, BPD leads to lower average repair bandwidth than previous piggybacking designs at the equal sub-packetization level. However, BPD needs larger fields to maintain the MDS property. We prove two upper bounds on the field size for explicit BPD and existential constructions, respectively. By computer search, specific BPD can be given over a field much smaller than the proved upper bounds. As examples, BPD codes with sub-packetization$l=4$based on the$[{12,8}]$and$[{14,10}]$Reed-Solomon codes are given over$\mathbb {F}_{2^{8}}$, which obtain about 16% savings in the average repair bandwidth over previous designs with$l\leq 4$.