On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage

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. We show the existence of Exact-Repair MDS codes that achieve minimum repair bandwidth (matching the cutset lower bound) for arbitrary admissible (n, k, d), i.e., k < n and k ≤ d ≤ n− 1. Our approach is based on interference alignment techniques and uses vector linear codes which allow to split symbols into arbitrarily small subsymbols.