An Outer Bound on the Storage-Bandwidth Tradeoff of Exact-Repair Cooperative Regenerating Codes

Abstract

Cooperative regenerating codes are a kind of erasure codes, which are optimal in terms of minimizing the repair bandwidth. An $(n,k,d,r)$ -cooperative regenerating code has $n$ storage nodes, where $k$ arbitrary nodes are enough to reconstruct original data, and $r$ failed nodes can be repaired cooperatively with the help of $d$ arbitrary surviving nodes. In the regenerating-code framework, there exists a tradeoff between the storage capacity of each node $\alpha $ and the repair bandwidth $\gamma $ , but the problem of specifying the optimal storage-bandwidth tradeoff of the exact-repair cooperative regenerating codes remains open. A key contribution of this paper is that an outer bound on the storage-bandwidth tradeoff of exact-repair linear cooperative regenerating codes is proposed. This result can be regarded as a generalization of the outer bound proposed by Prakash et al. , which specifies the optimal tradeoff of exact-repair regenerating codes for the case of $d=k=n-1$ . The proposed outer bound suggests the $(\alpha ,\gamma )$ pairs that no exact-repair codes can achieve but only functional-repair codes can. By observing the size of the set of such $(\alpha ,\gamma )$ pairs, the performance of the proposed outer bound is evaluated under various parameter settings.