Exact-repair MDS codes for distributed storage using interference alignment
The high repair cost of (n, k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address bandwidth-optimal (n, k, d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to arbitrary d survivor nodes, where k ≤ d ≤ n − 1. Under scalar-linear codes which do not permit symbol-splitting, we construct Exact-Repair MDS codes that are optimal in repair bandwidth for the case of k/n ≤ 1/2 and d ≥ 2k − 1. Our codes are deterministic and require a finite-field size of at most 2(n − k). Under vector-linear codes which allow for the break-up of stored symbols into arbitrarily small subsymbols, we show the existence of optimal Exact-Repair codes for the entire admissible range of possible (n, k, d), i.e., k < n and k ≤ d ≤ n − 1. That is, we establish the existence of vector-linear Exact-Repair MDS codes that match the fundamental cutset lower bound. Our approach for both the constructive scalar-linear code design and for the existence of vector-linear codes is based on interference alignment techniques.